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

(C) For a probability distribution,the sum of all probabilities must be equal to $1$. Given $P(X=k) = \frac{3^{ck}}{k!}$ for $k = 1, 2, 3, \ldots$,we

Vedclass · Accepted Answer

$\log_3(\log_e 2)$

Vedclass · Answer

(C) For a probability distribution,the sum of all probabilities must be equal to $1$. Given $P(X=k) = \frac{3^{ck}}{k!}$ for $k = 1, 2, 3, \ldots$,we have: $\sum_{k=1}^{\infty} P(X=k) = \sum_{k=1}^{\infty} \frac{(3^c)^k}{k!} = 1$. Recall the Taylor series expansion for the exponential function: $e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!} = 1 + \sum_{k=1}^{\infty} \frac{x^k}{k!}$. Therefore,$\sum_{k=1}^{\infty} \frac{x^k}{k!} = e^x - 1$. Substituting $x = 3^c$,we get: $e^{3^c} - 1 = 1$,which implies $e^{3^c} = 2$. Taking the natural logarithm on both sides: $3^c = \log_e 2$. Taking the logarithm base $3$ on both sides: $c = \log_3(\log_e 2)$. Thus,the correct option is $C$.

$X=x_i$	$-2$	$-1$	$0$	$1$	$2$
$P(X=x_i)$	$0.2$	$0.3$	$0.15$	$0.25$	$0.1$

If the probability function of a random variable $X$ is given by $P(X=k) = \frac{3^{ck}}{k!}$ for $k = 1, 2, 3, \ldots$ (where $c$ is a constant),then $c =$

Similar Questions

If a variable takes values $0, 1, 2, ..., n$ with frequencies proportional to the binomial coefficients $^nC_0, ^nC_1, ..., ^nC_n$,find the mean of the distribution.

In a game,a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decides to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he wins or loses. (in $/216$)

$A$ typist claims that he prepares a typed page with typo errors of $1$ per $10$ pages. In a typing assignment of $40$ pages, if the probability that the typo errors are at most $2$ is $p$, then $e^2 p=$

The value of $C$ for which $P(X = k) = Ck^2$ can serve as the probability function of a random variable $X$ that takes values $0, 1, 2, 3, 4$ is

If the probability distribution function of a random variable $X$ is given as follows:
$X=x_i$ $-2$ $-1$ $0$ $1$ $2$
$P(X=x_i)$ $0.2$ $0.3$ $0.15$ $0.25$ $0.1$

Then $F(0)$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module