If the probability function of a random varia | vedclass

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

Question

(A) Given the probability mass function $P(X=n) = \frac{k(n+1)}{3^n}$ for $n \in \{0, 1, 2, \dots\}$.Since the sum of all probabilities must be $1$,we

Vedclass · Accepted Answer

$\frac{20}{27}$

Vedclass · Answer

(A) Given the probability mass function $P(X=n) = \frac{k(n+1)}{3^n}$ for $n \in \{0, 1, 2, \dots\}$.Since the sum of all probabilities must be $1$,we have $\sum_{n=0}^{\infty} P(X=n) = 1$.$k \sum_{n=0}^{\infty} \frac{n+1}{3^n} = k \left( \frac{1}{3^0} + \frac{2}{3^1} + \frac{3}{3^2} + \frac{4}{3^3} + \dots \right) = 1$.This is an arithmetico-geometric series with $a=1$,$d=1$,and $r=\frac{1}{3}$.The sum of the series is $S = \frac{a}{1-r} + \frac{dr}{(1-r)^2} = \frac{1}{1-1/3} + \frac{1 \cdot (1/3)}{(1-1/3)^2} = \frac{3}{2} + \frac{1/3}{4/9} = \frac{3}{2} + \frac{3}{4} = \frac{9}{4}$.Thus,$k \cdot \frac{9}{4} = 1 \implies k = \frac{4}{9}$.We need to find $P(X $P(X=0) = k \cdot \frac{0+1}{3^0} = k \cdot 1 = \frac{4}{9}$.$P(X=1) = k \cdot \frac{1+1}{3^1} = k \cdot \frac{2}{3} = \frac{4}{9} \cdot \frac{2}{3} = \frac{8}{27}$.$P(X < 2) = \frac{4}{9} + \frac{8}{27} = \frac{12+8}{27} = \frac{20}{27}$.

If the probability function of a random variable $X$ is given by $P(X=n) = \frac{k(n+1)}{3^n}$ for $n \in \mathbb{N} \cup \{0\}$ where $k$ is a constant,then $P(X < 2) = $

Similar Questions

If $X$ is a Poisson variate with $P(X=0)=0.8$,then the variance of $X$ is

Bismuth has a half-life of $5$ days. If a sample originally has a mass of $800 \text{ mg}$, then the mass remaining after $30$ days will be: (in $\text{ mg}$)

If $X$ is a Poisson variate such that $\frac{5}{3} k = P(X=2) = P(X=3)$,then $P(X=5) =$

If the $c.d.f.$ (cumulative distribution function) is given by $F(x) = \frac{x-25}{10}$,then $P(27 \leq x \leq 33) = \_\_\_\_$

If the probability mass function (p.m.f.) of a random variable $X$ is $P(X=x) = \frac{1}{10}$ for $x = 1, 2, 3, \ldots, 10$,and $0$ otherwise,then $\operatorname{Var}(X)$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module