If P(X=x)=kleft(frac{3}{8}right)^{X}, x=1,2,3 | vedclass

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(C) Given that $P(X=x)=k\left(\frac{3}{8}\right)^X$ for $x=1, 2, 3, \ldots$ is a probability distribution function. Since the sum of all probabilities

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$\frac{5}{3}$

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(C) Given that $P(X=x)=k\left(\frac{3}{8}\right)^X$ for $x=1, 2, 3, \ldots$ is a probability distribution function. Since the sum of all probabilities must be equal to $1$,we have $\sum_{x=1}^{\infty} P(X=x) = 1$. Substituting the given expression: $k \sum_{x=1}^{\infty} \left(\frac{3}{8}\right)^x = 1$. This is an infinite geometric series with the first term $a = \frac{3}{8}$ and common ratio $r = \frac{3}{8}$. The sum of an infinite geometric series is given by $S = \frac{a}{1-r}$. Thus,$k \left( \frac{3/8}{1-3/8} \right) = 1$. $k \left( \frac{3/8}{5/8} \right) = 1$. $k \left( \frac{3}{5} \right) = 1$. Therefore,$k = \frac{5}{3}$.

$X=x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P(X=x)$	$0$	$k$	$2k$	$2k$	$3k$	$k^2$	$2k^2$	$7k^2+k$

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

$X = x$	$-1.5$	$-0.5$	$0.5$	$1.5$	$2.5$
$P(X = x)$	$0.05$	$0.2$	$0.15$	$0.25$	$0.35$

If $P(X=x)=k\left(\frac{3}{8}\right)^{X}, x=1,2,3, \ldots$ is the probability distribution function of a discrete random variable $X$,then $k=$

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