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

(C) $\frac{3}{4}$ and $\frac{1}{8}$

(A) $\frac{3}{4}$ and $\frac{1}{9}$

(B) $\frac{3}{4}$ and $\frac{1}{16}$

(D) $\frac{3}{8}$ and $\frac{1}{8}$

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Probability

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, \dots, \infty)

. Then the mean of the distribution and

P (X

is positive and even) respectively are :

Answer: (C)

The value

j = 0

contributes nothing to the mean, so

E (X) = j = 1 \sum \infty \frac{j}{3 ^{j}} = \frac{1/3}{( 1 - \frac{1}{3} ) ^{2}} = \frac{3}{4}

The positive even values are

j = 2, 4, 6, \dots

, and

P (X positive and even) = k = 1 \sum \infty \frac{1}{3 ^{2 k}} = \frac{1/9}{1 - \frac{1}{9}} = \frac{1}{8}

So the mean is

\frac{3}{4}

and the probability that

X

is positive and even is

\frac{1}{8}

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Single correct option