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Q.

Let X denote the number of vowels in word selected at random from this sentence “find the expected value and standard deviation of the random variable X” (Consider X as a word with one letter).

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a

$2, \sqrt{2}$

b

$\sqrt{2}, 2$

c

2, 2

d

$\sqrt{2}, \sqrt{2}$

answer is A.

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Detailed Solution

X :	0	1	2	3	4	5
P(X):	$\frac{1}{12}$	$\frac{5}{12}$	$\frac{2}{12}$	$\frac{2}{12}$	$\frac{1}{12}$	$\frac{1}{12}$

$E (x) = 0 (\frac{1}{12}) + 1 (\frac{5}{12}) + 2 (\frac{2}{12}) + 3 (\frac{2}{12}) + 4 (\frac{1}{12}) + 5 (\frac{1}{12}) = 2$

$variance = 0 (\frac{1}{12}) + 1 (\frac{5}{12}) + 4 (\frac{2}{12}) + 9 (\frac{2}{12}) + 16 (\frac{1}{12}) + 25 (\frac{1}{12}) - 2^{2} = 6 - 4 = 2$

$standard deviation = \sqrt{variance} = \sqrt{2}$

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Let X denote the number of vowels in word selected at random from this sentence “find the expected value and standard deviation of the random variable X” (Consider X as a word with one letter).