Let X be a random variable such that the probability function of … - Vidyadip

MCQ

Let $X$ be a random variable such that the probability function of a distribution is given by $P(X=$ 0) $=\frac{1}{2}, \mathrm{P}(\mathrm{X}=\mathrm{j})=\frac{1}{3^{j}}(\mathrm{j}=1,2,3, \ldots, \infty)$. Then the mean of the distribution and $\mathrm{P}(\mathrm{X}$ is positive and even) respectively are:

A
$\frac{3}{4}$ and $\frac{1}{9}$
B
$\frac{3}{4}$ and $\frac{1}{16}$
C
$\frac{3}{8}$ and $\frac{1}{8}$
✓
$\frac{3}{4}$ and $\frac{1}{8}$

✓

Answer

Correct option: D.

$\frac{3}{4}$ and $\frac{1}{8}$

d
mean $=\sum x_{i} p_{i}=\sum_{r=0}^{\infty} r \cdot \frac{1}{3^{r}}=\frac{3}{4}$

$p(x$ is even $)=\frac{1}{3^{2}}+\frac{1}{3^{4}}+\ldots \infty$

$=\frac{\frac{1}{9}}{1-\frac{1}{9}}=\frac{1}{8}$

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