Question:
Let $x$ be a random variable such that the probability function of a distribution is given by
$\mathrm{P}(\mathrm{X}=0)=\frac{1}{2}, \mathrm{P}(\mathrm{X}=\mathrm{j})=\frac{1}{3^{\mathrm{j}}}(\mathrm{j}=1,2,3, \ldots, \infty)$
Then the mean of the distribution and $\mathrm{P}(\mathrm{X}$ is positive and even) respectively are:
Correct Option: , 2
Solution:
mean $=\sum x_{i} p_{i}=\sum_{r=0}^{\infty} r \cdot \frac{1}{3^{r}}=\frac{3}{4}$
$\mathrm{p}(\mathrm{x}$ is even $)=\frac{1}{3^{2}}+\frac{1}{3^{4}}+\ldots \infty$
$=\frac{\frac{1}{9}}{1-\frac{1}{9}}=\frac{1 / 9}{8 / 9}=\frac{1}{8}$
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