Let the p.m.f. (probablity mass function) of random variable x be… - Vidyadip

Question

Let the p.m.f. (probablity mass function) of random variable $x$ be.
$\begin{array}{rlr}
P (x) & =\left(\frac{4}{x}\right)\left(\frac{5}{9}\right)^x\left(\frac{4}{9}\right)^{4-x}, & x=0,1,2,3,4 \\
& =0, & \text { otherwise }
\end{array}$
Find $E (x)$ and $\operatorname{Var}(x)$.

✓

Answer

$p(x)=\left(\frac{4}{x}\right)\left(\frac{5}{9}\right)^x\left(\frac{4}{9}\right)^{4-x}, x=0,1,2, \ldots, 4$
Comparing with $p ( x )=\left(\frac{ n }{ x }\right)( p )^{ x }( q )^{ n - x }$
$\therefore n =4, p =\frac{5}{9}, q =\frac{4}{9}$
$\begin{aligned} & E(x)=n p=4 \times \frac{5}{9}=\frac{20}{9} \\ & V(x)=n p q=4 \times \frac{5}{9} \times \frac{4}{9}=\frac{80}{81}\end{aligned}$

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