Question

Three numbers are selected at random (without replacement) from first six positive integers. Let X denotes the largest of the three numbers obtained. Find the probability distribution of X. Also, find the mean and variance of the distribution.

Answer 1

As the first six positive integers are 1, 2, 3, 4, 5, 6 so, the variate X takes values 3, 4, 5 and 6.

The probability distribution is given as :

$\begin{array}{ccccc}X& 3& 4& 5& 6\\ P\left(X\right)& \frac{1}{20}& \frac{3}{20}& \frac{6}{20}& \frac{10}{20}\end{array}$

Now mean, $\mu =\sum XP\left(X\right)=\frac{105}{20}or\frac{21}{4}$

And, variance = $=\sum X^{2}P\left(X\right)-{\mu }^2=\frac{63}{80}$.