Expectation and Variance
Discrete Case ($X$ with PMF $P(X=x)$)
| Quantity |
Formula |
| Mean / Expected Value |
$\mu = E(X) = \sum x , P(X = x)$ |
| Expectation of $g(X)$ |
$E[g(X)] = \sum g(x) , P(X = x)$ |
| $E(X^2)$ |
$E(X^2) = \sum x^2 , P(X = x)$ |
| Variance |
$\text{Var}(X) = \sigma^2 = \sum (x - \mu)^2 , P(X = x)$ |
| Computational variance |
$\text{Var}(X) = E(X^2) - [E(X)]^2$ |
| Standard deviation |
$\sigma = \sqrt{\text{Var}(X)}$ |
Continuous Case ($X$ with PDF $f(x)$)
| Quantity |
Formula |
| Mean / Expected Value |
$\mu = E(X) = \int_{-\infty}^{\infty} x,f(x),dx$ |
| Expectation of $g(X)$ |
$E[g(X)] = \int_{-\infty}^{\infty} g(x),f(x),dx$ |
| $E(X^2)$ |
$E(X^2) = \int_{-\infty}^{\infty} x^2,f(x),dx$ |
| Variance |
$\text{Var}(X) = \sigma^2 = \int_{-\infty}^{\infty} (x - \mu)^2,f(x),dx$ |
| Computational variance |
$\text{Var}(X) = \int_{-\infty}^{\infty} x^2,f(x),dx - \mu^2 = E(X^2) - [E(X)]^2$ |
| Standard deviation |
$\sigma = \sqrt{\text{Var}(X)}$ |
Linear Transformation Rules ($a$, $b$ constants)
| Rule |
Expectation |
Variance |
| Constant |
$E(a) = a$ |
$\text{Var}(a) = 0$ |
| Scaling |
$E(aX) = a,E(X)$ |
$\text{Var}(aX) = a^2,\text{Var}(X)$ |
| Shift + scale |
$E(aX + b) = a,E(X) + b$ |
$\text{Var}(aX + b) = a^2,\text{Var}(X)$ |
Adding $b$ shifts the mean but does not affect variance/spread.
Procedure
- Find $E(X)$: Multiply each $x$ by its probability (or $x f(x)$) and sum/integrate.
- Find $E(X^2)$: Multiply each $x^2$ by its probability (or $x^2 f(x)$) and sum/integrate.
- Compute variance: $\text{Var}(X) = E(X^2) - [E(X)]^2$.
- For $Y = aX + b$: Apply linear transformation rules above.
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