Let X be a random variable having a Binomial | vedclass

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(B) Given the Binomial distribution $B(n, p)$ with $n=7$,the probability mass function is $P(X=k) = {^nC_k} p^k q^{n-k}$,where $q = 1-p$. Given $P(X=3

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$\frac{35}{36}$

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(B) Given the Binomial distribution $B(n, p)$ with $n=7$,the probability mass function is $P(X=k) = {^nC_k} p^k q^{n-k}$,where $q = 1-p$. Given $P(X=3) = 5 P(X=4)$. Substituting the formula: ${^7C_3} p^3 q^4 = 5 	imes {^7C_4} p^4 q^3$ Since ${^7C_3} = {^7C_4} = 35$,we can divide both sides by $35 p^3 q^3$: $q = 5p$ Since $q = 1-p$,we have $1-p = 5p$,which implies $6p = 1$,so $p = \frac{1}{6}$. Then $q = 1 - \frac{1}{6} = \frac{5}{6}$. The variance of a Binomial distribution is given by $Var(X) = npq$. $Var(X) = 7 	imes \frac{1}{6} 	imes \frac{5}{6} = \frac{35}{36}$.

Let $X$ be a random variable having a Binomial distribution $B(7, p)$. If $P(X=3) = 5 P(X=4)$,then the variance of $X$ is:

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