Q.Find the variance and standard deviation of the random variable X whose probability distribution is given below: $x$: 0, 1, 2, 3; $P(X=x)$: $\dfrac{1}{8}, \dfrac{3}{8}, \dfrac{3}{8}, \dfrac{1}{8}$.

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 …