GATE DA 2024 | Question 11

MCQ · 1 mark · −0.33

Consider the following statements:

(i) The mean and variance of a Poisson random variable are equal.
(ii) For a standard normal random variable, the mean is zero and the variance is one.

Which ONE of the following options is correct?

  1. Option A: Both (i) and (ii) are true (correct)
  2. Option B: (i) is true and (ii) is false
  3. Option C: (ii) is true and (i) is false
  4. Option D: Both (i) and (ii) are false

Answer

(A)

Explanation

(i) A Poisson random variable with parameter \(\lambda\) has \(P(X = k) = \dfrac{e^{-\lambda}\lambda^{k}}{k!}\), and both its mean and its variance equal \(\lambda\). True.

(ii) The standard normal distribution \(N(0, 1)\) is, by definition, the normal distribution with mean 0 and variance 1; any normal variable is brought to it by \(Z = (X - \mu)/\sigma\). True.

Both statements hold.

Answer: A