ML-DA-law of total expectation.ipynb

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    "# ML, Data Analysis\n",
    "### Probability: law of total expectation \n",
    "\n",
    "The Law of Total Expectation is a fundamental rule in probability that relates the expected value of a random variable to its conditional expectations. For two random variables $X$ and $Y$, the law states that:\n",
    "<br>$E[X]=E[E[X|Y]]$\n",
    "<hr>\n",
    "\n",
    "For a **discrete random variable** $Y$, the law of total expectation becomes:\n",
    "<br>$E[X]=∑_y P(Y=y)\\cdot E[X|Y=y]$\n",
    "<br>where $P(Y=y)$ is the probability of the random variable $Y$ takes the value $y$. Moreover, $E[X|Y=y]$ is the expected value of $X$ given that $Y=y$.\n",
    "<hr>\n",
    "\n",
    "For a **continuous random variable** $Y$, the law of total expectation becomes:\n",
    "<br>$E[X]=\\int_{-\\infty}^{\\infty} E[X|Y=y]\\cdot f_Y(y)dy$\n",
    "<br>where $f_Y(y)$ is the probability (density) function (PDF) of $Y$.\n",
    "<hr>\n",
    "\n",
    "**Contents:**\n",
    " - Using the law of total expectation for a discrete random variable $Y$\n",
    " - Using the law of total expectation for a continuous random variable $Y$\n",
    "<hr>\n",
    "https://github.com/ostad-ai/Machine-Learning\n",
    "<br> Explanation: https://www.pinterest.com/HamedShahHosseini/Machine-Learning/background-knowledge"
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    "# Import thr required module\n",
    "import numpy as np\n",
    "# Import the required function for integration\n",
    "from scipy.integrate import quad"
   ]
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     "text": [
      "Expected number of ice creams sold a day: 36.5\n"
     ]
    }
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   "source": [
    "# Example: discrete random variable\n",
    "# Y represents the type of weather: sunny, cloudy, rainy, or snowy.\n",
    "# X represents the number of ice creams sold in a day, which depends on the weather.\n",
    "# We want to compute the expected number of ice creams sold E[X]\n",
    "\n",
    "# Define the probabilities of each weather type\n",
    "weather_probabilities = {\n",
    "    \"sunny\": 0.5,  # Probability of sunny weather\n",
    "    \"cloudy\": 0.3,  # Probability of cloudy weather\n",
    "    \"rainy\": 0.1,   # Probability of rainy weather\n",
    "    \"snowy\": 0.1    # Probability of snowy weather\n",
    "}\n",
    "\n",
    "# Define the conditional expectations E[X | Y = y] for each weather type\n",
    "conditional_expectations = {\n",
    "    \"sunny\": 55,  # Expected ice creams sold if sunny\n",
    "    \"cloudy\": 25, # Expected ice creams sold if cloudy\n",
    "    \"rainy\": 10,  # Expected ice creams sold if rainy\n",
    "    \"snowy\": 5    # Expected ice creams sold if snowy\n",
    "}\n",
    "\n",
    "# Compute the total expectation E[X] using the Law of Total Expectation\n",
    "total_expectation = 0\n",
    "for weather, prob in weather_probabilities.items():\n",
    "    total_expectation += prob * conditional_expectations[weather]\n",
    "\n",
    "# Print the result\n",
    "print(f\"Expected number of ice creams sold a day: {total_expectation}\")"
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     "text": [
      "Expected number of ice creams sold a day: 40.00000000382662\n"
     ]
    }
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    "# Example: continuous random variable\n",
    "\n",
    "# X is the number of ice creams sold, which depends on the temperature Y. \n",
    "# The relationship is given in function conditional_expectation.\n",
    "# Y is a continuous random variable representing the temperature\n",
    "# (in Celsius) on a given day, with a uniform distribution between 0 and 30.\n",
    "\n",
    "# Define the PDF of Y\n",
    "def pdf_Y(y):\n",
    "    return 1/30 if 0 <= y <= 30 else 0\n",
    "\n",
    "# Define the conditional expectation E[X | Y = y]\n",
    "def conditional_expectation(y):\n",
    "    return 10 + 2 * y\n",
    "\n",
    "# Define the integrand for the Law of Total Expectation\n",
    "def integrand(y):\n",
    "    return conditional_expectation(y) * pdf_Y(y)\n",
    "\n",
    "# Compute the expected value using numerical integration\n",
    "expected_value, _ = quad(integrand, -np.inf, np.inf)\n",
    "\n",
    "print(f\"Expected number of ice creams sold a day: {expected_value}\")"
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