Subtopic: Mean, median, mode and standard deviation
Question Statement:
Which of the following statements regarding probability distributions and statistical parameters are CORRECT?
Options:
(A) For a Poisson distribution with parameter $\lambda$, the mean and the variance are both equal to $\lambda$
(B) For any continuous random variable, the normal distribution is symmetric about its mean, with mean = median = mode
(C) If $X$ and $Y$ are independent random variables, then $\text{Var}(2X - 3Y) = 4\text{Var}(X) - 9\text{Var}(Y)$
(D) The standard normal distribution has mean equal to 0 and standard deviation equal to 1
Answer Key:
A, B, D
Step-by-Step Solution & Explanation
1. In a Poisson distribution, $E(X) = \lambda$ and $\text{Var}(X) = \lambda$. (Statement A is True)
2. A normal distribution is bell-shaped and symmetric about $\mu$, so mean = median = mode. (Statement B is True)
3. For independent random variables, $\text{Var}(aX + bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y)$. Thus $\text{Var}(2X - 3Y) = 4\text{Var}(X) + 9\text{Var}(Y)$. The minus sign makes Statement C False.
4. By definition, $Z \sim \mathcal{N}(0, 1)$ has mean $\mu = 0$ and $\sigma = 1$. (Statement D is True)