Two numbers are selected at random (without replacement) from fir… - Vidyadip

Question

Two numbers are selected at random (without replacement) from first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution.

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Answer

$\text{x}:$	2	3	4	5	6
$\text{P(x):}$	$\frac{1}{15}$	$\frac{2}{15}$	$\frac{3}{15}$	$\frac{4}{15}$	$\frac{5}{15}$
$\text{x.P(x):}$	$\frac{2}{15}$	$\frac{6}{15}$	$\frac{12}{15}$	$\frac{20}{15}$	$\frac{30}{15}$
$\text{x}^{2}\text{P(x):}$	$\frac{4}{15}$	$\frac{18}{15}$	$\frac{48}{15}$	$\frac{100}{15}$	$\frac{80}{15}$

$\text{Mean} = \sum \text{x. P(x)} = \frac{70}{15}=\frac{14}{3}$
$\text{Variance} = \sum \text{x}^{2}\text{p(x) = (Mean)}^{2} = \frac{350}{15}-\frac{196}{9} = \frac{14}{9} $

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