Q.A box contains $3$ defective and $7$ non-defective items. Two items are selected at random without replacement. Let $X$ denote the number of defective items selected. Find the probability distribution of $X$ and hence find its mean.

Mean and Variance of a Random Variable

A random variable $X$ assigns a number to each outcome, and its probability distribution lists each value $x_i$ with its probability $p_i$ (all $p_i\ge 0$ and $\sum p_i=1$).

Mean (expected value)

The mean $\mu=E(X)$ is the long-run average -- a weighted average of the values, each weighted by its probability:

For a function such as $X^2$, weight the function's values: $E(X^2)=\sum x_i^{\,2}\,p_i$.

Variance and standard deviation

Variance measures the spread about the mean. The working formula is:

and the standard deviation is $\sigma=\sqrt{\operatorname{Var}(X)}$ (back in the original units). …