Question

Let \(X\) be a random variable having binomial distribution \(B(7, p)\). If \(P(X = 3) = 5P(X = 4)\), then the sum of the mean and the variance of \(X\) is:

• A. \(\frac{105}{16}\)
• B. \(\frac{7}{16}\)
• C. \(\frac{77}{36}\)
• D. \(\frac{49}{16}\)

Hint:

For binomial distributions, use the formulas for mean \((np)\) and variance \((np(1-p))\) directly, and carefully simplify conditions involving probabilities.