A random variable X takes the values 0, 1, 2, | vedclass

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

Question

(A) The sum of all probabilities in a probability distribution must be equal to $1$. Therefore,$\sum_{x=0}^{\infty} P(X=x) = 1$. Substituting the give

Vedclass · Accepted Answer

$\frac{16}{25}$

Vedclass · Answer

(A) The sum of all probabilities in a probability distribution must be equal to $1$. Therefore,$\sum_{x=0}^{\infty} P(X=x) = 1$. Substituting the given expression: $\sum_{x=0}^{\infty} k(x+1)\left(\frac{1}{5}ight)^x = 1$. This is an arithmetico-geometric series of the form $\sum_{x=0}^{\infty} (x+1)r^x$ where $r = \frac{1}{5}$. The sum of the series $\sum_{x=0}^{\infty} (x+1)r^x = 1 + 2r + 3r^2 + \dots = \frac{1}{(1-r)^2}$. Substituting $r = \frac{1}{5}$: $\frac{1}{(1 - 1/5)^2} = \frac{1}{(4/5)^2} = \frac{1}{16/25} = \frac{25}{16}$. Thus,$k 	imes \frac{25}{16} = 1$,which gives $k = \frac{16}{25}$. We need to find $P(X=0)$. $P(X=0) = k(0+1)\left(\frac{1}{5}ight)^0 = k 	imes 1 	imes 1 = k$. Therefore,$P(X=0) = \frac{16}{25}$.

Outcome	Probability
$\omega_{1}$	$0.1$
$\omega_{2}$	$0.2$
$\omega_{3}$	$0.3$
$\omega_{4}$	$0.4$
$\omega_{5}$	$0.5$
$\omega_{6}$	$0.6$
$\omega_{7}$	$0.7$

$X=x$	$0$	$1$	$2$	$3$	$4$
$P(X=x)$	$\frac{1}{3}$	$\frac{1}{2}$	$0$	$\frac{1}{6}$	$0$

$A$ random variable $X$ takes the values $0, 1, 2, 3, \dots$ with probability $P(X=x) = k(x+1)\left(\frac{1}{5}\right)^x$,where $k$ is a constant. Then $P(X=0)$ is

Similar Questions

From a lot of $10$ items,which include $3$ defective items,a sample of $5$ items is drawn at random. Let the random variable $X$ denote the number of defective items in the sample. If the variance of $X$ is $\sigma^2$,then $96 \sigma^2$ is equal to....................

$A$ six-faced fair die is thrown until $2$ appears. What is the probability that $2$ appears in an even number of trials? (The die has six faces numbered $1, 2, 3, 4, 5,$ and $6$.)

$A$ $1$-rupee coin,a $2$-rupee coin,a $5$-rupee coin,and a $10$-rupee coin are tossed simultaneously. The expected value of the sum of the values of the coins that show heads up is:

If $m$ and ${\sigma ^2}$ are the mean and variance of a random variable $X$,whose distribution is given by:
$X=x$ $0$ $1$ $2$ $3$ $4$
$P(X=x)$ $\frac{1}{3}$ $\frac{1}{2}$ $0$ $\frac{1}{6}$ $0$

,then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module