## Probability Distributions: PMFs and PDFs

| **Formula Name** | **PMFs (Discrete Random Variable)** | **PDFs (Continuous Random Variable)** |
|:---------------------:|:-------------------------------------------------------------------------:|:----------------------------------------------------------------------------:|
| **Expectation** | $E[X] = \sum_x x \cdot p_X(x)$ | $E[X] = \int_{-\infty}^{\infty} x \cdot f_X(x) \, dx$ |
| **Description** | The expectation (or mean) is the sum of all possible values of $X$ weighted by their probabilities. It represents the average value of $X$. | The expectation (or mean) is the integral of $x$ times the PDF over all possible values of $x$. It represents the average value of $X$. |
| **Variance** | $Var(X) = \sum_x (x - E[X])^2 \cdot p_X(x)$ | $Var(X) = \int_{-\infty}^{\infty} (x - E[X])^2 \cdot f_X(x) \, dx$ |
| **Description** | The variance measures how much the values of $X$ deviate from the mean. It is the weighted average of the squared deviations from the mean. | The variance measures the spread of $X$ around its mean. It is the integral of the squared deviations from the mean, weighted by the PDF. |
| **Standard Deviation** | $SD(X) = \sqrt{Var(X)}$ | $SD(X) = \sqrt{Var(X)}$ |
| **Description** | The standard deviation is the square root of the variance. It provides a measure of spread in the same units as the random variable itself. | The standard deviation is the square root of the variance and measures the spread of $X$ around its mean in the same units as $X$. |