Question

If \( X \sim B(n, p) \) then \( \frac{P(X=k){P(X=k-1)} = \)}

• A. \( \frac{n-k}{k-1} \cdot \frac{p}{q} \)
• B. \( \frac{n-k+1}{k+1} \cdot \frac{p}{q} \)
• C. \( \frac{n+1}{k} \cdot \frac{q}{p} \)
• D. \( \frac{n-k+1}{k} \cdot \frac{p}{q} \)

Hint:

This ratio is useful for finding the mode of a binomial distribution.

Answer 1

The Correct Option is D

Step 1: Concept Binomial probability formula: \(P(X=k) = \binom{n}{k} p^k q^{n-k}\).

Step 2: Meaning We need to simplify the ratio of two consecutive terms of the binomial distribution.

Step 3: Analysis Ratio = \(\frac{\binom{n}{k} p^k q^{n-k}}{\binom{n}{k-1} p^{k-1} q^{n-k+1}}\) \(= \frac{\binom{n}{k}}{\binom{n}{k-1}} \cdot \frac{p}{q}\) \(= \frac{n!}{k!(n-k)!} \cdot \frac{(k-1)!(n-k+1)!}{n!} \cdot \frac{p}{q}\) \(= \frac{n-k+1}{k} \cdot \frac{p}{q}\).

Step 4: Conclusion The simplified ratio is \(\frac{n-k+1}{k} \cdot \frac{p}{q}\). Final Answer: (D)