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

Answers (1)

Total number of ways of selecting 3 numbers out of first 6 positive integers $\mathrm{ = {^6C_3} = 20}$

As X: the smallest of the three numbers selected

so, X = 1,2,3,4

X	1	2	3	4
P(X)	$\mathrm{\frac{10}{20}}$	$\mathrm{\frac{6}{20}}$	$\mathrm{\frac{3}{20}}$	$\mathrm{\frac{1}{20}}$
XP(X)	$\mathrm{\frac{10}{20}}$	$\mathrm{\frac{12}{20}}$	$\mathrm{\frac{9}{20}}$	$\mathrm{\frac{4}{20}}$
X²P(X)	$\mathrm{\frac{10}{20}}$	$\mathrm{\frac{24}{20}}$	$\mathrm{\frac{27}{20}}$	$\mathrm{\frac{16}{20}}$

$\mathrm{Mean = \sum XP(X) = \frac{35}{20}\ or \ \frac{7}{4}}$

$\mathrm{Variance = \sum X^2P(X) - (Mean)^2 = \frac{77}{20}\ - \frac{49}{16}}$

$\mathrm{ = \frac{63}{80}}$

Posted by

Ravindra Pindel

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