{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "230760979d82fbcd",
   "metadata": {},
   "source": [
    "# Random Number Generation\n",
    "\n",
    "Random number generation is fundamental to machine learning and computational neuroscience. **BrainState** provides a comprehensive random number generation system built on JAX that combines:\n",
    "\n",
    "- 🎲 **NumPy-like API**: Familiar interface for easy adoption\n",
    "- 🔐 **Reproducibility**: Deterministic pseudorandom sequences\n",
    "- ⚡ **High Performance**: JIT-compiled random functions\n",
    "- 🔄 **Stateful Management**: Automatic key splitting and state handling\n",
    "\n",
    "By the end of this tutorial, you will understand:\n",
    "\n",
    "- How to generate random numbers with BrainState\n",
    "- The `RandomState` class and seed management\n",
    "- Available probability distributions\n",
    "- Random sampling and shuffling operations\n",
    "- Best practices for reproducible experiments"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "7724835268dc9f6",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:07.034881Z",
     "start_time": "2025-10-10T10:05:04.842551Z"
    },
    "execution": {
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     "shell.execute_reply": "2026-05-30T16:21:07.779003Z"
    }
   },
   "outputs": [],
   "source": [
    "import brainstate\n",
    "import numpy as np\n",
    "import jax.numpy as jnp\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "why_random",
   "metadata": {},
   "source": [
    "## Why BrainState Random?\n",
    "\n",
    "### JAX's Challenge\n",
    "\n",
    "JAX uses **explicit random keys** for reproducibility and parallelization:\n",
    "\n",
    "```python\n",
    "# JAX approach - manual key management\n",
    "key = jax.random.PRNGKey(0)\n",
    "key, subkey = jax.random.split(key)  # Must split manually\n",
    "x = jax.random.normal(subkey, (100,))\n",
    "```\n",
    "\n",
    "This is powerful but verbose. **BrainState** simplifies this:\n",
    "\n",
    "```python\n",
    "# BrainState approach - automatic key management\n",
    "brainstate.random.seed(0)\n",
    "x = brainstate.random.normal(0, 1, (100,))  # Keys handled automatically\n",
    "```\n",
    "\n",
    "### Key Features\n",
    "\n",
    "✅ **Automatic key splitting**: No manual key management  \n",
    "✅ **NumPy compatibility**: Drop-in replacement for `numpy.random`  \n",
    "✅ **Stateful interface**: Natural imperative programming style  \n",
    "✅ **JIT compatible**: Full performance with JAX transformations  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff4331f95839e85e",
   "metadata": {},
   "source": [
    "## API Overview\n",
    "\n",
    "The `brainstate.random` module provides a wide range of functions for random number generation and sampling from probability distributions.\n",
    "\n",
    "| Category | API Functions |\n",
    "|----------|---------------|\n",
    "| **Random Sampling** | `rand`, `randn`, `randint`, `random`, `choice` |\n",
    "| **Continuous Distributions** | `normal`, `uniform`, `beta`, `gamma`, `exponential`, `gumbel`, `laplace`, `logistic` |\n",
    "| **Discrete Distributions** | `binomial`, `poisson`, `bernoulli`, `categorical` |\n",
    "| **Shuffling** | `shuffle`, `permutation` |\n",
    "| **Seed Management** | `seed`, `get_key`, `set_key`, `split_key` |\n",
    "| **State Management** | `RandomState`, `DEFAULT` |\n",
    "\n",
    "### The DEFAULT RandomState\n",
    "\n",
    "All random functions in `brainstate.random` use a global `DEFAULT` instance of `RandomState`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "6a3746e719381c04",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:07.044513Z",
     "start_time": "2025-10-10T10:05:07.039504Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:07.782428Z",
     "iopub.status.busy": "2026-05-30T16:21:07.781984Z",
     "iopub.status.idle": "2026-05-30T16:21:07.813147Z",
     "shell.execute_reply": "2026-05-30T16:21:07.812232Z"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "An NVIDIA GPU may be present on this machine, but a CUDA-enabled jaxlib is not installed. Falling back to cpu.\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Default RandomState:\n",
      "RandomState(Array((), dtype=key<fry>) overlaying:\n",
      "[2325272776 1178154024])\n",
      "\n",
      "Type: <class 'brainstate.random.RandomState'>\n"
     ]
    }
   ],
   "source": [
    "# The DEFAULT random state\n",
    "print(\"Default RandomState:\")\n",
    "print(brainstate.random.DEFAULT)\n",
    "print(f\"\\nType: {type(brainstate.random.DEFAULT)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "seed_management",
   "metadata": {},
   "source": [
    "## Seed Management and Reproducibility\n",
    "\n",
    "### Setting Seeds\n",
    "\n",
    "For reproducible experiments, always set a seed at the beginning:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "5480509c4a1c58b1",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:07.373445Z",
     "start_time": "2025-10-10T10:05:07.045892Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:07.814844Z",
     "iopub.status.busy": "2026-05-30T16:21:07.814706Z",
     "iopub.status.idle": "2026-05-30T16:21:08.136077Z",
     "shell.execute_reply": "2026-05-30T16:21:08.135319Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Seed set to 42\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "First sequence: [0.72766423 0.78786755 0.18169427 0.26263022 0.11072934]\n",
      "Second sequence: [0.72766423 0.78786755 0.18169427 0.26263022 0.11072934]\n",
      "\n",
      "Identical? True\n"
     ]
    }
   ],
   "source": [
    "# Set seed for reproducibility\n",
    "brainstate.random.seed(42)\n",
    "print(\"Seed set to 42\")\n",
    "\n",
    "# Generate some numbers\n",
    "x1 = brainstate.random.rand(5)\n",
    "print(f\"First sequence: {x1}\")\n",
    "\n",
    "# Reset to same seed\n",
    "brainstate.random.seed(42)\n",
    "x2 = brainstate.random.rand(5)\n",
    "print(f\"Second sequence: {x2}\")\n",
    "print(f\"\\nIdentical? {jnp.allclose(x1, x2)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "key_management",
   "metadata": {},
   "source": [
    "### Advanced: Key Management\n",
    "\n",
    "For advanced use cases, you can directly access and manipulate keys:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "27dc45c891e1bb31",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:07.515452Z",
     "start_time": "2025-10-10T10:05:07.377286Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:08.137750Z",
     "iopub.status.busy": "2026-05-30T16:21:08.137495Z",
     "iopub.status.idle": "2026-05-30T16:21:08.311331Z",
     "shell.execute_reply": "2026-05-30T16:21:08.310155Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Current key: Array((), dtype=key<fry>) overlaying:\n",
      "[1832780943  270669613]\n",
      "\n",
      "Split into 4 keys:\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  Key 0: Array((), dtype=key<fry>) overlaying:\n",
      "[1705926158  899080142]\n",
      "  Key 1: Array((), dtype=key<fry>) overlaying:\n",
      "[4095997477  317277840]\n",
      "  Key 2: Array((), dtype=key<fry>) overlaying:\n",
      "[1820970612 3729538270]\n",
      "  Key 3: Array((), dtype=key<fry>) overlaying:\n",
      "[3985668294 1940518238]\n"
     ]
    }
   ],
   "source": [
    "# Get current key\n",
    "current_key = brainstate.random.get_key()\n",
    "print(f\"Current key: {current_key}\")\n",
    "\n",
    "# Split key for parallel operations\n",
    "keys = brainstate.random.split_key(n=4)\n",
    "print(f\"\\nSplit into {len(keys)} keys:\")\n",
    "for i, key in enumerate(keys):\n",
    "    print(f\"  Key {i}: {key}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "df9f1b1a891a8113",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:07.692298Z",
     "start_time": "2025-10-10T10:05:07.554441Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:08.313677Z",
     "iopub.status.busy": "2026-05-30T16:21:08.313402Z",
     "iopub.status.idle": "2026-05-30T16:21:08.461298Z",
     "shell.execute_reply": "2026-05-30T16:21:08.460460Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "v1: [-0.39489815 -0.42596528 -1.0719717 ]\n",
      "v2: [ 0.6733885  -0.50893074 -0.28400496]\n",
      "v3 (restored): [-0.39489815 -0.42596528 -1.0719717 ]\n",
      "\n",
      "v1 == v3? True\n"
     ]
    }
   ],
   "source": [
    "# Save and restore state\n",
    "saved_key = brainstate.random.get_key()\n",
    "v1 = brainstate.random.randn(3)\n",
    "\n",
    "# Generate more numbers\n",
    "v2 = brainstate.random.randn(3)\n",
    "\n",
    "# Restore state\n",
    "brainstate.random.set_key(saved_key)\n",
    "v3 = brainstate.random.randn(3)\n",
    "\n",
    "print(f\"v1: {v1}\")\n",
    "print(f\"v2: {v2}\")\n",
    "print(f\"v3 (restored): {v3}\")\n",
    "print(f\"\\nv1 == v3? {jnp.allclose(v1, v3)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4a13411b1c9cfa2",
   "metadata": {},
   "source": [
    "## Random Sampling\n",
    "\n",
    "### Basic Sampling Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "3bdf5c4c909f4e3e",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:07.703526Z",
     "start_time": "2025-10-10T10:05:07.697100Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:08.463666Z",
     "iopub.status.busy": "2026-05-30T16:21:08.463386Z",
     "iopub.status.idle": "2026-05-30T16:21:08.468744Z",
     "shell.execute_reply": "2026-05-30T16:21:08.468001Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Uniform [0,1) samples: [0.72766423 0.78786755 0.18169427 0.26263022 0.11072934]\n"
     ]
    }
   ],
   "source": [
    "# Uniform [0, 1)\n",
    "brainstate.random.seed(42)\n",
    "uniform_samples = brainstate.random.rand(5)\n",
    "print(\"Uniform [0,1) samples:\", uniform_samples)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "938ca61e0dc053ce",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:07.822287Z",
     "start_time": "2025-10-10T10:05:07.726946Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:08.470361Z",
     "iopub.status.busy": "2026-05-30T16:21:08.470181Z",
     "iopub.status.idle": "2026-05-30T16:21:08.573909Z",
     "shell.execute_reply": "2026-05-30T16:21:08.572886Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Standard normal samples: [-0.21089035 -1.3627948  -0.04500385 -1.1536394   1.9141139 ]\n"
     ]
    }
   ],
   "source": [
    "# Standard normal (mean=0, std=1)\n",
    "normal_samples = brainstate.random.randn(5)\n",
    "print(\"Standard normal samples:\", normal_samples)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "93592d115567567a",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:07.965019Z",
     "start_time": "2025-10-10T10:05:07.827297Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:08.576236Z",
     "iopub.status.busy": "2026-05-30T16:21:08.576045Z",
     "iopub.status.idle": "2026-05-30T16:21:08.715234Z",
     "shell.execute_reply": "2026-05-30T16:21:08.714358Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Random integers [0,10): [0 2 0 0 8]\n"
     ]
    }
   ],
   "source": [
    "# Random integers\n",
    "int_samples = brainstate.random.randint(0, 10, size=5)\n",
    "print(\"Random integers [0,10):\", int_samples)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "sampling_ops",
   "metadata": {},
   "source": [
    "### Choice and Permutation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "589aa4e95bf0ab87",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:08.120406Z",
     "start_time": "2025-10-10T10:05:07.982011Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:08.717776Z",
     "iopub.status.busy": "2026-05-30T16:21:08.717616Z",
     "iopub.status.idle": "2026-05-30T16:21:08.866860Z",
     "shell.execute_reply": "2026-05-30T16:21:08.866086Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Random choices from [10 20 30 40 50]:\n",
      "[40 50 20]\n"
     ]
    }
   ],
   "source": [
    "# Random choice from array\n",
    "options = jnp.array([10, 20, 30, 40, 50])\n",
    "choices = brainstate.random.choice(options, size=3, replace=False)\n",
    "print(f\"Random choices from {options}:\")\n",
    "print(choices)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "a1b5ceb209d46dfe",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:08.268654Z",
     "start_time": "2025-10-10T10:05:08.140985Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:08.868253Z",
     "iopub.status.busy": "2026-05-30T16:21:08.868102Z",
     "iopub.status.idle": "2026-05-30T16:21:09.013023Z",
     "shell.execute_reply": "2026-05-30T16:21:09.012154Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Original: [0 1 2 3 4 5 6 7 8 9]\n",
      "Shuffled: [7 4 8 2 5 6 3 0 1 9]\n"
     ]
    }
   ],
   "source": [
    "# Shuffle array\n",
    "array = jnp.arange(10)\n",
    "shuffled = brainstate.random.shuffle(array)\n",
    "print(f\"Original: {array}\")\n",
    "print(f\"Shuffled: {shuffled}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "permutation_example",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:08.280850Z",
     "start_time": "2025-10-10T10:05:08.274766Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:09.015627Z",
     "iopub.status.busy": "2026-05-30T16:21:09.015381Z",
     "iopub.status.idle": "2026-05-30T16:21:09.021111Z",
     "shell.execute_reply": "2026-05-30T16:21:09.020206Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Random permutation of 0-9: [2 6 9 7 4 3 1 0 5 8]\n"
     ]
    }
   ],
   "source": [
    "# Random permutation\n",
    "perm = brainstate.random.permutation(10)\n",
    "print(f\"Random permutation of 0-9: {perm}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f574683a3cf8bd24",
   "metadata": {},
   "source": [
    "## Probability Distributions\n",
    "\n",
    "### Continuous Distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "2a4016c97e9ec69f",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:08.636784Z",
     "start_time": "2025-10-10T10:05:08.293170Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:09.022905Z",
     "iopub.status.busy": "2026-05-30T16:21:09.022744Z",
     "iopub.status.idle": "2026-05-30T16:21:09.285216Z",
     "shell.execute_reply": "2026-05-30T16:21:09.284342Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Normal(μ=5, σ=2):\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  Mean: 4.941 (expected: 5.0)\n",
      "  Std:  2.099 (expected: 2.0)\n"
     ]
    }
   ],
   "source": [
    "# Normal distribution\n",
    "brainstate.random.seed(0)\n",
    "normal_samples = brainstate.random.normal(loc=5.0, scale=2.0, size=1000)\n",
    "print(f\"Normal(μ=5, σ=2):\")\n",
    "print(f\"  Mean: {jnp.mean(normal_samples):.3f} (expected: 5.0)\")\n",
    "print(f\"  Std:  {jnp.std(normal_samples):.3f} (expected: 2.0)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "ad42cfc68efeb896",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:08.778700Z",
     "start_time": "2025-10-10T10:05:08.638503Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:09.287820Z",
     "iopub.status.busy": "2026-05-30T16:21:09.287552Z",
     "iopub.status.idle": "2026-05-30T16:21:09.475422Z",
     "shell.execute_reply": "2026-05-30T16:21:09.474648Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Uniform[0, 10):\n",
      "  Mean: 5.071 (expected: 5.0)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  Min:  0.012\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  Max:  9.953\n"
     ]
    }
   ],
   "source": [
    "# Uniform distribution\n",
    "uniform_samples = brainstate.random.uniform(low=0, high=10, size=1000)\n",
    "print(f\"\\nUniform[0, 10):\")\n",
    "print(f\"  Mean: {jnp.mean(uniform_samples):.3f} (expected: 5.0)\")\n",
    "print(f\"  Min:  {jnp.min(uniform_samples):.3f}\")\n",
    "print(f\"  Max:  {jnp.max(uniform_samples):.3f}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "more_distributions",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:09.503643Z",
     "start_time": "2025-10-10T10:05:08.783136Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:09.477039Z",
     "iopub.status.busy": "2026-05-30T16:21:09.476882Z",
     "iopub.status.idle": "2026-05-30T16:21:10.208110Z",
     "shell.execute_reply": "2026-05-30T16:21:10.207273Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Exponential(λ=0.5):\n",
      "  Mean: 2.021 (expected: 2.0)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Beta(α=2, β=5):\n",
      "  Mean: 0.292 (expected: 0.286)\n"
     ]
    }
   ],
   "source": [
    "# Exponential distribution\n",
    "exp_samples = brainstate.random.exponential(scale=2.0, size=1000)\n",
    "print(f\"\\nExponential(λ=0.5):\")\n",
    "print(f\"  Mean: {jnp.mean(exp_samples):.3f} (expected: 2.0)\")\n",
    "\n",
    "# Beta distribution\n",
    "beta_samples = brainstate.random.beta(a=2.0, b=5.0, size=1000)\n",
    "print(f\"\\nBeta(α=2, β=5):\")\n",
    "print(f\"  Mean: {jnp.mean(beta_samples):.3f} (expected: {2/(2+5):.3f})\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "discrete_dists",
   "metadata": {},
   "source": [
    "### Discrete Distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "22e71b4dd19744a2",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:10.072089Z",
     "start_time": "2025-10-10T10:05:09.521978Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:10.210460Z",
     "iopub.status.busy": "2026-05-30T16:21:10.210155Z",
     "iopub.status.idle": "2026-05-30T16:21:10.776940Z",
     "shell.execute_reply": "2026-05-30T16:21:10.776134Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Binomial(n=10, p=0.5):\n",
      "  Mean: 5.024 (expected: 5.0)\n",
      "  Std:  1.588 (expected: 1.581)\n"
     ]
    }
   ],
   "source": [
    "# Binomial distribution\n",
    "binomial_samples = brainstate.random.binomial(n=10, p=0.5, size=1000)\n",
    "print(f\"Binomial(n=10, p=0.5):\")\n",
    "print(f\"  Mean: {jnp.mean(binomial_samples):.3f} (expected: 5.0)\")\n",
    "print(f\"  Std:  {jnp.std(binomial_samples):.3f} (expected: {jnp.sqrt(10*0.5*0.5):.3f})\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "df476ddc47d413a9",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:10.424088Z",
     "start_time": "2025-10-10T10:05:10.086874Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:10.778930Z",
     "iopub.status.busy": "2026-05-30T16:21:10.778762Z",
     "iopub.status.idle": "2026-05-30T16:21:11.176293Z",
     "shell.execute_reply": "2026-05-30T16:21:11.175216Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Poisson(λ=3):\n",
      "  Mean: 2.979 (expected: 3.0)\n",
      "  Var:  2.949 (expected: 3.0)\n"
     ]
    }
   ],
   "source": [
    "# Poisson distribution\n",
    "poisson_samples = brainstate.random.poisson(lam=3.0, size=1000)\n",
    "print(f\"\\nPoisson(λ=3):\")\n",
    "print(f\"  Mean: {jnp.mean(poisson_samples):.3f} (expected: 3.0)\")\n",
    "print(f\"  Var:  {jnp.var(poisson_samples):.3f} (expected: 3.0)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "visualization",
   "metadata": {},
   "source": [
    "### Visualizing Distributions\n",
    "\n",
    "Let's visualize several distributions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "viz_distributions",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:12.705426Z",
     "start_time": "2025-10-10T10:05:10.439396Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:11.179307Z",
     "iopub.status.busy": "2026-05-30T16:21:11.179134Z",
     "iopub.status.idle": "2026-05-30T16:21:11.179403Z",
     "shell.execute_reply": "2026-05-30T16:21:11.178445Z"
    }
   },
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 1500x1000 with 6 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "brainstate.random.seed(42)\n",
    "\n",
    "fig, axes = plt.subplots(2, 3, figsize=(15, 10))\n",
    "\n",
    "# Normal\n",
    "samples = brainstate.random.normal(0, 1, 10000)\n",
    "axes[0, 0].hist(np.array(samples), bins=50, density=True, alpha=0.7, edgecolor='black')\n",
    "axes[0, 0].set_title('Normal(μ=0, σ=1)', fontsize=12, fontweight='bold')\n",
    "axes[0, 0].set_xlabel('Value')\n",
    "axes[0, 0].set_ylabel('Density')\n",
    "axes[0, 0].grid(alpha=0.3)\n",
    "\n",
    "# Uniform\n",
    "samples = brainstate.random.uniform(0, 1, 10000)\n",
    "axes[0, 1].hist(np.array(samples), bins=50, density=True, alpha=0.7, edgecolor='black')\n",
    "axes[0, 1].set_title('Uniform(0, 1)', fontsize=12, fontweight='bold')\n",
    "axes[0, 1].set_xlabel('Value')\n",
    "axes[0, 1].set_ylabel('Density')\n",
    "axes[0, 1].grid(alpha=0.3)\n",
    "\n",
    "# Exponential\n",
    "samples = brainstate.random.exponential(1.0, 10000)\n",
    "axes[0, 2].hist(np.array(samples), bins=50, density=True, alpha=0.7, edgecolor='black')\n",
    "axes[0, 2].set_title('Exponential(λ=1)', fontsize=12, fontweight='bold')\n",
    "axes[0, 2].set_xlabel('Value')\n",
    "axes[0, 2].set_ylabel('Density')\n",
    "axes[0, 2].grid(alpha=0.3)\n",
    "\n",
    "# Beta\n",
    "samples = brainstate.random.beta(2, 5, 10000)\n",
    "axes[1, 0].hist(np.array(samples), bins=50, density=True, alpha=0.7, edgecolor='black')\n",
    "axes[1, 0].set_title('Beta(α=2, β=5)', fontsize=12, fontweight='bold')\n",
    "axes[1, 0].set_xlabel('Value')\n",
    "axes[1, 0].set_ylabel('Density')\n",
    "axes[1, 0].grid(alpha=0.3)\n",
    "\n",
    "# Binomial\n",
    "samples = brainstate.random.binomial(20, 0.5, 10000)\n",
    "axes[1, 1].hist(np.array(samples), bins=21, density=True, alpha=0.7, edgecolor='black')\n",
    "axes[1, 1].set_title('Binomial(n=20, p=0.5)', fontsize=12, fontweight='bold')\n",
    "axes[1, 1].set_xlabel('Value')\n",
    "axes[1, 1].set_ylabel('Density')\n",
    "axes[1, 1].grid(alpha=0.3)\n",
    "\n",
    "# Poisson\n",
    "samples = brainstate.random.poisson(5, 10000)\n",
    "axes[1, 2].hist(np.array(samples), bins=range(0, 20), density=True, alpha=0.7, edgecolor='black')\n",
    "axes[1, 2].set_title('Poisson(λ=5)', fontsize=12, fontweight='bold')\n",
    "axes[1, 2].set_xlabel('Value')\n",
    "axes[1, 2].set_ylabel('Density')\n",
    "axes[1, 2].grid(alpha=0.3)\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "891ef892ae243f2d",
   "metadata": {},
   "source": [
    "## Custom RandomState Instances\n",
    "\n",
    "For advanced use cases, you can create custom `RandomState` instances with independent random streams:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "cf8892d7a4a23211",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:44.097556Z",
     "start_time": "2025-10-10T10:05:44.091292Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:11.181506Z",
     "iopub.status.busy": "2026-05-30T16:21:11.181315Z",
     "iopub.status.idle": "2026-05-30T16:21:11.187079Z",
     "shell.execute_reply": "2026-05-30T16:21:11.186060Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RNG 1:\n",
      "RandomState(Array((), dtype=key<fry>) overlaying:\n",
      "[ 0 42])\n"
     ]
    }
   ],
   "source": [
    "# Create custom random generators\n",
    "rng1 = brainstate.random.RandomState(42)\n",
    "rng2 = brainstate.random.RandomState(123)\n",
    "\n",
    "print(\"RNG 1:\")\n",
    "print(rng1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "64ebd13e567cc95d",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:44.845564Z",
     "start_time": "2025-10-10T10:05:44.838882Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:11.188944Z",
     "iopub.status.busy": "2026-05-30T16:21:11.188782Z",
     "iopub.status.idle": "2026-05-30T16:21:11.195311Z",
     "shell.execute_reply": "2026-05-30T16:21:11.194663Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Samples from RNG1: [ 0.60576403  0.7990441  -0.908927   -0.63525754 -1.2226585 ]\n",
      "Samples from RNG2: [-0.7828054  -1.5373377  -0.5513038  -0.02385257  1.164293  ]\n",
      "\n",
      "Are they different? True\n"
     ]
    }
   ],
   "source": [
    "# Generate independent random sequences\n",
    "samples1 = rng1.randn(5)\n",
    "samples2 = rng2.randn(5)\n",
    "\n",
    "print(f\"Samples from RNG1: {samples1}\")\n",
    "print(f\"Samples from RNG2: {samples2}\")\n",
    "print(f\"\\nAre they different? {not jnp.allclose(samples1, samples2)}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "9c1f8d3c3a5a1fe",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:45.490800Z",
     "start_time": "2025-10-10T10:05:45.486129Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:11.197905Z",
     "iopub.status.busy": "2026-05-30T16:21:11.197478Z",
     "iopub.status.idle": "2026-05-30T16:21:11.203379Z",
     "shell.execute_reply": "2026-05-30T16:21:11.202601Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "After re-seeding RNG1 to 999: [ 0.8200472  -0.35510033 -1.2458946  -0.33925015 -0.7285477 ]\n"
     ]
    }
   ],
   "source": [
    "# Re-seed custom RNG\n",
    "rng1.seed(999)\n",
    "samples3 = rng1.randn(5)\n",
    "print(f\"\\nAfter re-seeding RNG1 to 999: {samples3}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "practical_examples",
   "metadata": {},
   "source": [
    "## Practical Examples\n",
    "\n",
    "### Example 1: Mini-Batch Sampling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "minibatch_example",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:05:47.892862Z",
     "start_time": "2025-10-10T10:05:47.074855Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:11.205750Z",
     "iopub.status.busy": "2026-05-30T16:21:11.205549Z",
     "iopub.status.idle": "2026-05-30T16:21:12.605449Z",
     "shell.execute_reply": "2026-05-30T16:21:12.604686Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Created 4 mini-batches\n",
      "First batch shape: X=(32, 10), y=(32,)\n"
     ]
    }
   ],
   "source": [
    "def create_mini_batches(X, y, batch_size=32, shuffle=True):\n",
    "    \"\"\"Create mini-batches for training.\"\"\"\n",
    "    n_samples = len(X)\n",
    "    \n",
    "    # Shuffle indices\n",
    "    if shuffle:\n",
    "        indices = brainstate.random.permutation(n_samples)\n",
    "    else:\n",
    "        indices = jnp.arange(n_samples)\n",
    "    \n",
    "    # Create batches\n",
    "    batches = []\n",
    "    for start_idx in range(0, n_samples, batch_size):\n",
    "        end_idx = min(start_idx + batch_size, n_samples)\n",
    "        batch_indices = indices[start_idx:end_idx]\n",
    "        batches.append((X[batch_indices], y[batch_indices]))\n",
    "    \n",
    "    return batches\n",
    "\n",
    "# Generate dummy dataset\n",
    "brainstate.random.seed(0)\n",
    "X = brainstate.random.randn(100, 10)  # 100 samples, 10 features\n",
    "y = brainstate.random.randint(0, 2, 100)  # Binary labels\n",
    "\n",
    "# Create mini-batches\n",
    "batches = create_mini_batches(X, y, batch_size=32)\n",
    "print(f\"Created {len(batches)} mini-batches\")\n",
    "print(f\"First batch shape: X={batches[0][0].shape}, y={batches[0][1].shape}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dropout_example",
   "metadata": {},
   "source": [
    "### Example 2: Dropout Layer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "dropout_impl",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:06:28.590054Z",
     "start_time": "2025-10-10T10:06:28.518870Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:12.607701Z",
     "iopub.status.busy": "2026-05-30T16:21:12.607469Z",
     "iopub.status.idle": "2026-05-30T16:21:12.692532Z",
     "shell.execute_reply": "2026-05-30T16:21:12.691556Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Input:  [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]\n",
      "Output: [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]\n",
      "Zeros:  0/10\n"
     ]
    }
   ],
   "source": [
    "class Dropout(brainstate.nn.Module):\n",
    "    \"\"\"Dropout layer with random masking.\"\"\"\n",
    "    \n",
    "    def __init__(self, drop_rate=0.5):\n",
    "        super().__init__()\n",
    "        self.drop_rate = drop_rate\n",
    "\n",
    "    def __call__(self, x):\n",
    "        fit = brainstate.environ.get('fit', False)\n",
    "        if not fit:\n",
    "            return x\n",
    "        \n",
    "        # Generate random mask\n",
    "        keep_prob = 1.0 - self.drop_rate\n",
    "        mask = brainstate.random.bernoulli(keep_prob, x.shape)\n",
    "        \n",
    "        # Apply dropout and scale\n",
    "        return x * mask / keep_prob\n",
    "\n",
    "# Test dropout\n",
    "brainstate.random.seed(42)\n",
    "dropout = Dropout(drop_rate=0.3)\n",
    "\n",
    "x = jnp.ones(10)\n",
    "y = dropout(x)\n",
    "\n",
    "print(f\"Input:  {x}\")\n",
    "print(f\"Output: {y}\")\n",
    "print(f\"Zeros:  {jnp.sum(y == 0)}/10\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "noise_example",
   "metadata": {},
   "source": [
    "### Example 3: Noisy Neural Network"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "noisy_network",
   "metadata": {
    "ExecuteTime": {
     "end_time": "2025-10-10T10:06:35.063528Z",
     "start_time": "2025-10-10T10:06:34.811438Z"
    },
    "execution": {
     "iopub.execute_input": "2026-05-30T16:21:12.694991Z",
     "iopub.status.busy": "2026-05-30T16:21:12.694751Z",
     "iopub.status.idle": "2026-05-30T16:21:13.052713Z",
     "shell.execute_reply": "2026-05-30T16:21:13.051728Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Output 1: [-0.29487485 -0.37535232 -0.19899406]\n",
      "Output 2: [-0.30561778 -0.33523202 -0.16594572]\n",
      "Difference: [-0.01074293  0.0401203   0.03304833]\n"
     ]
    }
   ],
   "source": [
    "class NoisyLayer(brainstate.nn.Module):\n",
    "    \"\"\"Linear layer with Gaussian noise.\"\"\"\n",
    "    \n",
    "    def __init__(self, d_in, d_out, noise_std=0.1):\n",
    "        super().__init__()\n",
    "        self.noise_std = noise_std\n",
    "        \n",
    "        # Parameters with noise\n",
    "        self.w = brainstate.ParamState(brainstate.random.randn(d_in, d_out) * 0.1)\n",
    "        self.b = brainstate.ParamState(jnp.zeros(d_out))\n",
    "    \n",
    "    def __call__(self, x):\n",
    "        # Add weight noise\n",
    "        w_noisy = self.w.value + brainstate.random.normal(0, self.noise_std, self.w.value.shape)\n",
    "        \n",
    "        # Forward pass\n",
    "        return x @ w_noisy + self.b.value\n",
    "\n",
    "# Create and test\n",
    "brainstate.random.seed(0)\n",
    "layer = NoisyLayer(5, 3, noise_std=0.01)\n",
    "\n",
    "x = jnp.ones(5)\n",
    "y1 = layer(x)\n",
    "y2 = layer(x)  # Different due to noise\n",
    "\n",
    "print(f\"Output 1: {y1}\")\n",
    "print(f\"Output 2: {y2}\")\n",
    "print(f\"Difference: {y2 - y1}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "best_practices",
   "metadata": {},
   "source": [
    "## Best Practices\n",
    "\n",
    "### 1. Always Set Seeds for Reproducibility\n",
    "\n",
    "```python\n",
    "# At the start of your script/notebook\n",
    "brainstate.random.seed(42)\n",
    "```\n",
    "\n",
    "### 2. Use Custom RNGs for Independent Streams\n",
    "\n",
    "```python\n",
    "# For data augmentation\n",
    "aug_rng = brainstate.random.RandomState(seed=123)\n",
    "\n",
    "# For model initialization  \n",
    "init_rng = brainstate.random.RandomState(seed=456)\n",
    "```\n",
    "\n",
    "### 3. Save and Restore State for Checkpoints\n",
    "\n",
    "```python\n",
    "# Save state\n",
    "checkpoint = {\n",
    "    'model': model.state_dict(),\n",
    "    'rng_key': brainstate.random.get_key()\n",
    "}\n",
    "\n",
    "# Restore state\n",
    "brainstate.random.set_key(checkpoint['rng_key'])\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "summary",
   "metadata": {},
   "source": [
    "## Summary\n",
    "\n",
    "In this tutorial, you learned:\n",
    "\n",
    "✅ **Seed management** for reproducible experiments  \n",
    "✅ **Random sampling** with various distributions  \n",
    "✅ **Custom RandomState** instances for independent streams  \n",
    "✅ **Practical applications** in neural networks  \n",
    "✅ **Best practices** for random number generation  \n",
    "\n",
    "### Key Points\n",
    "\n",
    "- 🎲 BrainState provides a NumPy-like interface to JAX's random system\n",
    "- 🔐 Always set seeds for reproducibility\n",
    "- ⚡ Random functions are JIT-compatible and high-performance\n",
    "- 🔄 State is automatically managed for you\n",
    "\n",
    "### Next Steps\n",
    "\n",
    "Continue with:\n",
    "- **Neural Network Modules** - Use random initialization in models\n",
    "- **Program Transformations** - Combine randomness with JIT, vmap, and grad\n",
    "- **Training Loops** - Implement stochastic gradient descent\n",
    "\n",
    "### Additional Resources\n",
    "\n",
    "- 📚 [JAX Random Documentation](https://jax.readthedocs.io/en/latest/jax.random.html)\n",
    "- 🎲 [NumPy Random Documentation](https://numpy.org/doc/stable/reference/random/index.html)\n",
    "- 🧠 [BrainState API Reference](https://brainx.chaobrain.com/brainstate/)"
   ]
  }
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