Question

Two numbers are selected at random (without replacement) from the first five positive integers.

Let X denote the larger of the two numbers obtained. Find the mean and variance of X.

Answer 1

Since X denotes the larger of the two numbers obtained from 1,2,3,4 and 5.

So values of X :2,3,4,5.

$\begin{array}{cc}\text{X}& \text{P(X)}\\ 2& 2\times \frac{1}{5}\times \frac{1}{4}=\frac{2}{20}\\ 3& 4\times \frac{1}{5}\times \frac{1}{4}=\frac{4}{20}\\ 4& 6\times \frac{1}{5}\times \frac{1}{4}=\frac{6}{20}\\ 5& 8\times \frac{1}{5}\times \frac{1}{4}=\frac{8}{20}\end{array}$

Now, mean =$\sum XP\left(X\right)=2\times \frac{2}{20}+3\times \frac{4}{20}+4\times \frac{6}{20}+5\times \frac{8}{20}=\frac{4+12+24+40}{20}=4$

And, variance =$\sum X^{2}P\left(X\right)-[Mean{]}^2=2^2\times \frac{2}{20}+3^2\times \frac{4}{20}+4^2\times \frac{6}{20}+5^2\times \frac{8}{20}-(4{)}^2\Rightarrow =\frac{8+36+96+200}{20}-16=17-16=1$