Random variables & distributions

Japanese school year: University year 1

What you learn

Learn about random variables whose values are determined by trial outcomes, their probability distributions, and how to compute expectation and variance. These concepts quantify central tendency and dispersion, forming the foundation for statistical inference. Prior understanding of basic probability rules and sequence summations is required.

Go to practice

Key points

Use this formula to calculate the expected value (mean) $E(X)$ of a discrete random variable $X$ . Multiply each possible value $x_i$ by its probability $p_i$ and sum them all up.

E(X) = \sum_{i=1}^{n} x_i p_i

Use this formula to efficiently compute the spread (variance $V(X)$ ) of random variable $X$ . Subtract the square of the mean $(E(X))^2$ from the expected value of the squared variable $E(X^2)$ .

V(X) = E(X^2) - (E(X))^2

Use this formula to find the new expected value when random variable $X$ is multiplied by $a$ and shifted by $b$ . The new mean is simply $a$ times the original expectation $E(X)$ , plus $b$ .

E(aX + b) = aE(X) + b

This formula shows how scaling and shifting affect variance. Multiplying $X$ by $a$ scales the variance by $a^2$ , whereas adding a constant $b$ does not change the spread at all.

V(aX + b) = a^2 V(X)

Choose a set to practice.

Practice 1Coming soon