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

Given that the probability of a man hitting a target with a gun is $\frac{1}{3}$. If he fires $8$ times,then the probability of his hitting the target at least twice is

$A$ die is thrown $100$ times. The standard deviation of getting an even number is:

The probability that a person who undergoes a certain operation will survive is $0.2$. If $5$ patients undergo similar operations,then the probability that exactly four will survive is

$A$ rifleman is firing at a distant target and has only $10 \%$ chance of hitting it. The least number of rounds he must fire to have more than $50 \%$ chance of hitting it at least once, is

The centers for disease control have determined that when a person is given a vaccine, the probability that the person will develop immunity to a virus is $0.8$. If $8$ people are given this vaccine, then the probability that exactly $4$ will develop immunity is...

Vedclass Test Series

Exam Paper Generator

Online Exam Module