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If $P (X = x) = c {(\frac{2}{3})}^{x}; x = 1, 2, 3, 4, \dots$ is a probability distribution function of a random variable X, then the value of c is

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$\frac{1}{4}$

$\frac{1}{3}$

$\frac{1}{2}$

$\frac{1}{6}$

answer is C.

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Detailed Solution

$P (X = x) = c {(\frac{2}{3})}^{x}; x = 1, 2, 3, 4, \dots$

$\Rightarrow c (\frac{2}{3}) + c {(\frac{2}{3})}^{2} + \cdot \cdot \cdot$

$\Rightarrow C (\frac{2}{3}) [1 + \frac{2}{3} + \cdot \cdot \cdot]$

$a = 1, r = 2 / 3$

$\Rightarrow S_{\infty} = \frac{a}{1 - r}$

$C (\frac{2}{3}) \times \frac{1}{1 - 2 / 3} = 1$

$c = \frac{1}{2}$

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