If X is a random variable with probability di | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) For a probability distribution,the sum of all probabilities must be $1$: $\sum_{k=0}^{\infty} P(X=k) = 1$. Substituting the given expression: $c \

Vedclass · Accepted Answer

$\frac{1}{18}$

Vedclass · Answer

(B) For a probability distribution,the sum of all probabilities must be $1$: $\sum_{k=0}^{\infty} P(X=k) = 1$. Substituting the given expression: $c \sum_{k=0}^{\infty} \frac{2k+3}{3^k} = 1$. Let $S = \sum_{k=0}^{\infty} \frac{2k+3}{3^k} = 3 \sum_{k=0}^{\infty} \frac{1}{3^k} + 2 \sum_{k=0}^{\infty} \frac{k}{3^k}$. The first part is a geometric series: $3 	imes \frac{1}{1 - 1/3} = 3 	imes \frac{3}{2} = \frac{9}{2}$. The second part is an arithmetico-geometric series: $\sum_{k=0}^{\infty} k x^k = \frac{x}{(1-x)^2}$. For $x = 1/3$,this is $\frac{1/3}{(1-1/3)^2} = \frac{1/3}{4/9} = \frac{3}{4}$. So,$S = \frac{9}{2} + 2 	imes \frac{3}{4} = \frac{9}{2} + \frac{3}{2} = 6$. Since $c 	imes S = 1$,we have $c = 1/6$. Now,$P(X=3) = \frac{(2(3)+3)c}{3^3} = \frac{9c}{27} = \frac{c}{3}$. Substituting $c = 1/6$,we get $P(X=3) = \frac{1/6}{3} = \frac{1}{18}$.

If $X$ is a random variable with probability distribution $P(X=k) = \frac{(2k+3)c}{3^k}$,$k=0, 1, 2, \ldots, \infty$,then $P(X=3) =$

Similar Questions

$A$ person who tosses an unbiased coin gains two points for turning up a head and loses one point for a tail. If three coins are tossed and the total score $X$ is observed,then the range of $X$ is

Suppose the number of accidents occurring on a highway in each day follows a Poisson random variable with parameter $3$. Then,what is the probability that no accidents occur today?

For the following probability distribution,the standard deviation of the random variable $X$ is:

$X$ $2$ $3$ $4$

$P(X=x)$ $0.2$ $0.5$ $0.3$

In a city,$10$ accidents take place in a span of $50$ days. Assuming that the number of accidents follows the Poisson distribution,the probability that three or more accidents occur in a day is:

If $X$ is a Poisson variate such that $P(X=1) = 2P(X=2)$,then $P(X=3)$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module