Subtopic: Poisson, normal and binomial distributions
Question Statement:
For a standard normal distribution $Z \sim N(0, 1)$, the skewness and kurtosis are respectively:
Options:
(A) $0$ and $3$
(B) $1$ and $3$
(C) $0$ and $0$
(D) $0$ and $1$
Answer Key:
A
Step-by-Step Solution & Explanation
For any normal distribution:
1. Symmetry implies that the third central moment is zero, so skewness $\gamma_1 = \frac{\mu_3}{\sigma^3} = 0$.
2. The fourth central moment is $\mu_4 = 3\sigma^4$, so kurtosis $\beta_2 = \frac{\mu_4}{\sigma^4} = 3$ (and excess kurtosis is $\beta_2 - 3 = 0$).
Thus, option A is correct.