Question

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

• A. $\frac{3}{8}$ and $\frac{1}{8}$
• B. $\frac{3}{4}$ and $\frac{1}{8}$
• C. $\frac{3}{4}$ and $\frac{1}{9}$
• D. $\frac{3}{4}$ and $\frac{1}{16}$

Answer 1

Answer: (B)

mean $=\sum x_{1} p_{i}=\displaystyle\sum_{r=0}^{\infty} r \cdot \frac{1}{3^{r}}=\frac{3}{4}$
$p(x$ is even $)=\frac{1}{3^{2}}+\frac{1}{3^{4}}+\ldots . \infty$
$=\frac{\frac{1}{9}}{1-\frac{1}{9}}=\frac{1 / 9}{8 / 9}=\frac{1}{8}$