Let X sim B(n, p). If E(X)=5 and operatorname | vedclass

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(B) Given that $X$ follows a binomial distribution $B(n, p)$.For a binomial distribution,the mean $E(X) = np = 5$ and the variance $\operatorname{Var}

Vedclass · Accepted Answer

$\left(\frac{1}{2}\right)^{10}$

Vedclass · Answer

(B) Given that $X$ follows a binomial distribution $B(n, p)$.For a binomial distribution,the mean $E(X) = np = 5$ and the variance $\operatorname{Var}(X) = npq = 2.5$.Dividing the variance by the mean,we get $q = \frac{npq}{np} = \frac{2.5}{5} = 0.5 = \frac{1}{2}$.Since $p + q = 1$,we have $p = 1 - 0.5 = 0.5 = \frac{1}{2}$.Substituting $p = \frac{1}{2}$ into $np = 5$,we get $n 	imes \frac{1}{2} = 5$,which implies $n = 10$.We need to find $P(X The probability mass function is given by $P(X = k) = {}^{n}C_{k} p^{k} q^{n-k}$.For $k = 0$,$P(X = 0) = {}^{10}C_{0} \left(\frac{1}{2}ight)^{0} \left(\frac{1}{2}ight)^{10-0} = 1 	imes 1 	imes \left(\frac{1}{2}ight)^{10} = \left(\frac{1}{2}ight)^{10}$.

Let $X \sim B(n, p)$. If $E(X)=5$ and $\operatorname{Var}(X)=2.5$,then find $P(X < 1)$.

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