Which of the following statements are correct ?
(A) In a binomial distribution, Mean = np, Variance = npq, where n = number of trials, p = probability of success and q = probability of failure.
(B) In a Poisson distribution, Mean = Variance
(C) The standard normal probability curve is symmetric about mean.
(D) \( \mathrm{Var}\,(\mathrm{X})=\left[\mathrm{E}(\mathrm{X})\right]^{2}-\mathrm{E}\left(\mathrm{X}^{2}\right) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B), and (C) only
The standard results are:
Binomial: mean = np, variance = npq, with q = 1 − p
Poisson with parameter λ: mean = variance = λ
\( \displaystyle \mathrm{Var}(\mathrm{X})=\mathrm{E}(\mathrm{X}^{2})-\left[\mathrm{E}(\mathrm{X})\right]^{2} \)
So the binomial mean and variance are np and npq, the Poisson mean and variance are both λ, the standard normal curve is symmetric about its mean 0, and the variance is E(X2) minus the square of E(X).
Checking the statements
(A) In a binomial distribution, Mean = np, Variance = npq, where n = number of trials, p = probability of success and q = probability of failure. — Correct. These are the standard results.
(B) In a Poisson distribution, Mean = Variance — Correct. Both equal λ.
(C) The standard normal probability curve is symmetric about mean. — Correct. The curve is bell-shaped and symmetric about its mean, 0.
(D) \( \mathrm{Var}\,(\mathrm{X})=\left[\mathrm{E}(\mathrm{X})\right]^{2}-\mathrm{E}\left(\mathrm{X}^{2}\right) \) — Incorrect. The terms are the wrong way round; this is the negative of the variance.