Which of the following statements are correct?
(A) The mean of a binomial distribution is always greater than the variance.
(B) The mean of a Poisson distribution is always greater than the variance.
(C) If \( f(x) \) is the probability density function of a random variable \( X \), then \( P(a\le X\le b)=\int_a^b\displaystyle f(x)\,dx \).
(D) If \( X \) is a random variable and \( a,b \) are real numbers, then \( Var(aX+b)=a^2Var(X) \).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A),(C) and (D) only
For a binomial distribution, mean = np and variance = npq, where q = 1 − p < 1. For a Poisson distribution, mean = variance = λ.
So the binomial variance npq is less than the mean np, the Poisson variance equals the mean, \( P(a\le X\le b) \) is the integral of f(x) from a to b, and \( Var(aX+b)=a^2Var(X) \).
Checking the statements
(A) The mean of a binomial distribution is always greater than the variance. — Correct. npq < np because q < 1.
(B) The mean of a Poisson distribution is always greater than the variance. — Incorrect. Both the mean and the variance are λ, so they are equal.
(C) If \( f(x) \) is the probability density function of a random variable \( X \), then \( P(a\le X\le b)=\int_a^b\displaystyle f(x)\,dx \). — Correct. This is how probabilities are found from a density function.
(D) If \( X \) is a random variable and \( a,b \) are real numbers, then \( Var(aX+b)=a^2Var(X) \). — Correct. Adding b does not change the spread; multiplying by a multiplies the variance by \( a^2 \).