Q.Consider a random variable X with probability density function, $f(x)=\begin{cases}4x^3, & \text{if } 0<x<1 \\ 0, & \text{otherwise}\end{cases}$ Find $E(X)$ and $V(X)$.

The variance of a random variable $X$, written $Var(X)$ or $\sigma_x^2$, measures how spread out the values of $X$ are around its mean $\mu = E(X)$; it is defined as $Var(X) = \sum (x_i - \mu)^2 p_i$, the probability-weighted average of the squared deviations from the mean. In practice it is usually computed with the shortcut $Var(X) = E(X^2) - [E(X)]^2$, where $E(X^2) = \sum x_i^2 p_i$. The non-negative square root of the variance, $\sigma_x = \sqrt{Var(X)}$, is called the standard deviation, and is preferr …