Let a random variable X have a Binomial distr | vedclass

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

Question

(D) Let $X \sim B(n, p)$.Given that the mean $np = 8$ and variance $npq = 4$.Since $q = 1 - p$,we have $8q = 4$,which implies $q = \frac{1}{2}$ and $p

Vedclass · Accepted Answer

$137$

Vedclass · Answer

(D) Let $X \sim B(n, p)$.Given that the mean $np = 8$ and variance $npq = 4$.Since $q = 1 - p$,we have $8q = 4$,which implies $q = \frac{1}{2}$ and $p = \frac{1}{2}$.Substituting $p = \frac{1}{2}$ into $np = 8$,we get $n = 16$.We need to find $P(X \leq 2) = P(X=0) + P(X=1) + P(X=2)$.Using the formula $P(X=r) = {}^{n}C_{r} p^{r} q^{n-r}$,we have:$P(X \leq 2) = {}^{16}C_{0} (\frac{1}{2})^{0} (\frac{1}{2})^{16} + {}^{16}C_{1} (\frac{1}{2})^{1} (\frac{1}{2})^{15} + {}^{16}C_{2} (\frac{1}{2})^{2} (\frac{1}{2})^{14}$.$P(X \leq 2) = \frac{{}^{16}C_{0} + {}^{16}C_{1} + {}^{16}C_{2}}{2^{16}}$.Calculating the combinations: ${}^{16}C_{0} = 1$,${}^{16}C_{1} = 16$,and ${}^{16}C_{2} = \frac{16 	imes 15}{2} = 120$.Thus,$P(X \leq 2) = \frac{1 + 16 + 120}{2^{16}} = \frac{137}{2^{16}}$.Comparing this with $\frac{K}{2^{16}}$,we get $K = 137$.

Let a random variable $X$ have a Binomial distribution with mean $8$ and variance $4$. If $P(X \leq 2) = \frac{K}{2^{16}}$,then $K$ is

Similar Questions

$A$ rifle man is firing at a distant target and has only $10\%$ chance of hitting it. The minimum number of rounds he must fire in order to have $50\%$ chance of hitting it at least once is

The sum and product of the mean and variance of a binomial distribution are $82.5$ and $1350$ respectively. The number of trials in the binomial distribution is:

$A$ student appeared in an examination consisting of $8$ true-false type questions. The student guesses the answers with equal probability. The smallest value of $n$,so that the probability of guessing at least $n$ correct answers is less than $\frac{1}{2}$,is:

If $X \sim B\left(8, \frac{1}{2}\right)$,then $P(|X-4| \leq 2)=$

$A$ dice is thrown $5$ times,then the probability that an even number will come up exactly $3$ times is

Vedclass Test Series

Exam Paper Generator

Online Exam Module