Q.A commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station. Let $X$ denote the amount of time, in minutes, that the student waits for the train from the time he reaches the train station. It is known that the probability density function of $X$ is

[!FORMULA] $f(x)=\begin{cases}\dfrac1{30} & 0<x<30\\ 0 & \text{elsewhere}\end{cases}$

Obtain and interpret the expected value of the random variable $X$.

Mean (Definition 11.8): for a random variable $X$ with pmf/pdf $f(x)$,

$E(X)$ generalises the plain numerical average, weighting each value by its true probability rather than by $\tfrac1n$; it need not be a value $X$ can actually take, and is best read as the long-run average over many repetitions. Theorem 11.3 extends this to any function $g(X)$: $E(g(X))=\sum_x g(x)f(x)$ or $\int g(x)f(x)\,dx$; taking $g(X)=X^k$ gives the $k$-th moment $E(X^k)$.

Variance (Definition 11.9): $V(X)=E\big((X-E(X))^2\big)$, with the far more usable computing form

Standard deviation is $\sigma=\sqrt{V(X)}$; both are always $\ge0$. A smaller $\sigma^2$ means values cluster tightly around the mean; a larger $\sigma^2$ means they scatter more widely — even distributions sharing the same mean can differ sharply here. …