If X is a Poisson variate such that alpha = P | vedclass

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

Question

(B) Given that $X$ is a Poisson variate with parameter $\lambda$, the probability mass function is $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$.Given

Vedclass · Accepted Answer

$\frac{\alpha}{3}$

Vedclass · Answer

(B) Given that $X$ is a Poisson variate with parameter $\lambda$, the probability mass function is $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$.Given $\alpha = P(X=1) = P(X=2)$:$\frac{e^{-\lambda} \lambda^1}{1!} = \frac{e^{-\lambda} \lambda^2}{2!}$$\lambda = \frac{\lambda^2}{2} \Rightarrow \lambda = 2$ (since $\lambda > 0$).Now, calculate $\alpha$:$\alpha = P(X=1) = \frac{e^{-2} 	imes 2^1}{1!} = 2e^{-2}$.We need to find $P(X=4)$:$P(X=4) = \frac{e^{-\lambda} \lambda^4}{4!} = \frac{e^{-2} 	imes 2^4}{24} = \frac{16 e^{-2}}{24} = \frac{2}{3} e^{-2}$.Since $\alpha = 2e^{-2}$, we have $e^{-2} = \frac{\alpha}{2}$.Substituting this into the expression for $P(X=4)$:$P(X=4) = \frac{2}{3} 	imes \frac{\alpha}{2} = \frac{\alpha}{3}$.

$X$	$0$	$1$	$2$	$3$	$4$
$P(X=x)$	$2k$	$k$	$2k$	$4k$	$k$

If $X$ is a Poisson variate such that $\alpha = P(X=1) = P(X=2)$, then $P(X=4)$ is equal to

Similar Questions

In a Poisson distribution, if $P(X = 2)$ is twice $P(X = 1)$, then the standard deviation of the distribution is:

Two bad eggs are mixed accidentally with $10$ good ones. If three eggs are drawn at random from this lot in succession without replacement, then the variance of the probability distribution of the number of bad eggs drawn is

In a book of $500$ pages, it is found that there are $250$ typing errors. Assume that Poisson law holds for the number of errors per page. Then, the probability that a random sample of $2$ pages will contain no error, is:

If a die is thrown at random,then the expectation of the number on it is

The probability distribution of a discrete random variable $X$ is given by the table below:
$X$ $0$ $1$ $2$ $3$ $4$
$P(X=x)$ $2k$ $k$ $2k$ $4k$ $k$

If $a = P(X < 3)$ and $b = P(2 < X < 4)$,then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module