Question

The range of a random variable $X$ is $1,2,3,4....$
and the probabilities are given by $p(X=k)=\frac{c^k}{k!}k=1,2,3,\mathrm{4....},$, then the value of $C$ is

• A. $22$
• B. $\log e$
• C. ${\log }_{e}2$
• D. 4

Answer 1

Answer: (C)

${\log }_{e}2$

We know, $\sum P\left(X\right)=1$
$\Rightarrow c+\frac{c^2}{2!}+\frac{c^3}{3!}+...........=1$
$\Rightarrow 1+c+\frac{c^2}{2!}+\frac{c^3}{3!}+...........=1+1=2$
$\Rightarrow e^c=2\Rightarrow c={\log }_{e}2$