The value of the constant c,so that P(x)=clef | vedclass

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(B) For a discrete random variable $X$,the sum of all probabilities must be equal to $1$,i.e.,$\sum P(x) = 1$. Given $P(x) = c\left(\frac{2}{3}\right)

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$\frac{1}{2}$

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(B) For a discrete random variable $X$,the sum of all probabilities must be equal to $1$,i.e.,$\sum P(x) = 1$. Given $P(x) = c\left(\frac{2}{3}\right)^x$ for $x = 1, 2, 3, \ldots$. Therefore,$\sum_{x=1}^{\infty} c\left(\frac{2}{3}\right)^x = 1$. $c \left[ \frac{2}{3} + \left(\frac{2}{3}\right)^2 + \left(\frac{2}{3}\right)^3 + \ldots \right] = 1$. The expression inside the bracket is an infinite geometric series with first term $a = \frac{2}{3}$ and common ratio $r = \frac{2}{3}$. The sum of an infinite geometric series is given by $S = \frac{a}{1-r}$. $c \left[ \frac{2/3}{1 - 2/3} \right] = 1$. $c \left[ \frac{2/3}{1/3} \right] = 1$. $c(2) = 1$. $c = \frac{1}{2}$.

$X = x$	$1$	$2$	$3$	$4$	$5$	$6$
$P(X = x)$	$k$	$0$	$2k$	$5k$	$k$	$3k$

$X = x_{i}$	$-2$	$-1$	$0$	$1$	$2$	$3$
$P(X = x_{i})$	$0.1$	$k$	$0.2$	$2k$	$3k$	$k$

The value of the constant $c$,so that $P(x)=c\left(\frac{2}{3}\right)^{x}$,$x=1,2,3, \ldots$ is the probability distribution function of a discrete random variable $X$ is

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