Let X be a random variable such that the probability function of a distribution is given by P(X=0)=(1)/(2) , P(X=j)=frac13^j (j=1,2,3,ldots,infty) . Then the mean of the distribution and P(Xtext is positive and even) respectively are :

Probability: Let X be a random variable such that the probability function of a… | JEE Main 2021

JEE Main 2021 (Online) 25th July Evening ShiftQuestion 1

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 $P(X\text{ is positive and even})$ respectively are :

Probability

$\frac{3}{4}$ and $\frac{1}{8}$✓ Correct answer$\frac{3}{4}$ and $\frac{1}{16}$$\frac{3}{8}$ and $\frac{1}{8}$$\frac{3}{4}$ and $\frac{1}{9}$

Solution

Mean $=\sum_{j\ge1}\frac{j}{3^j}=\frac{1/3}{(1-1/3)^2}=\frac34$. $P(\text{positive even})=\sum_{m\ge1}\frac1{9^m}=\frac{1/9}{8/9}=\frac18$. Option (A).

Probability - JEE Mathematics question (JEE Main 2021 (Online) 25th July Evening Shift)

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