If X sim B(n, p), then the value of frac{P(X | vedclass

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(A) The probability mass function of a binomial distribution $X \sim B(n, p)$ is given by $P(X = k) = \binom{n}{k} p^k q^{n-k}$, where $q = 1 - p$.We

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$(\frac{n - k + 1}{k}) \frac{p}{q}$

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(A) The probability mass function of a binomial distribution $X \sim B(n, p)$ is given by $P(X = k) = \binom{n}{k} p^k q^{n-k}$, where $q = 1 - p$.We have $P(X = k) = \frac{n!}{k!(n-k)!} p^k q^{n-k}$ and $P(X = k-1) = \frac{n!}{(k-1)!(n-k+1)!} p^{k-1} q^{n-k+1}$.Taking the ratio: $\frac{P(X = k)}{P(X = k - 1)} = \frac{n!}{k!(n-k)!} p^k q^{n-k} 	imes \frac{(k-1)!(n-k+1)!}{n! p^{k-1} q^{n-k+1}}$.Simplifying the terms: $\frac{P(X = k)}{P(X = k - 1)} = \frac{(k-1)!}{k!} 	imes \frac{(n-k+1)!}{(n-k)!} 	imes \frac{p^k}{p^{k-1}} 	imes \frac{q^{n-k}}{q^{n-k+1}}$.This results in $\frac{1}{k} 	imes (n-k+1) 	imes p 	imes \frac{1}{q} = \frac{n-k+1}{k} \cdot \frac{p}{q}$.

If $X \sim B(n, p)$, then the value of $\frac{P(X = k)}{P(X = k - 1)}$ is

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