Random variables, mean & variance

Japanese school year: Math B

What you learn

This topic explores random variables whose values are determined by trial outcomes, calculating their expected value and variance. These measures are crucial for quantifying uncertainty and analyzing statistical risks in science and finance. Prior familiarity with foundational probability concepts and basic summation is recommended for this study.

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Key points

This formula calculates the expected value (mean) of a random variable $X$ . By multiplying each possible value $x_i$ by its probability $p_i$ and adding them together, you can find the long-term average outcome.

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

This formula calculates the variance $V(X)$ , which measures how widely values are spread out. You can find it quickly by subtracting the square of the mean, $(E(X))^2$ , from the mean of squared values, $E(X^2)$ .

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

This formula gives the standard deviation $\sigma(X)$ by taking the square root of the variance $V(X)$ . This returns the squared unit back to the original unit, making it easy to compare with the average.

\sigma(X) = \sqrt{V(X)}

This formula shows how the variance changes when $X$ is multiplied by $a$ and shifted by $b$ . Adding a constant $b$ does not change the spread, while scaling by $a$ multiplies the variance by $a^2$ .

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

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Practice 1Coming soon