Let \( X \sim N(0,1) \) be a standard normal random variable. We are asked to compute the raw moments:

\[ E[X^n] \quad \text{for } n = 1, 2, 3, 4 \]

Moment Definitions

For a random variable \( X \), the \( n \)-th raw moment is:

\[ \mu_n’ = E[X^n] \]

For the standard normal, the moments can be derived either by integrating against the density or using known properties of the distribution.

Computations

1. First Moment

\[ E[X] = 0 \]

(because the normal is symmetric about zero)

2. Second Moment (Variance)

\[ E[X^2] = \text{Var}(X) + (E[X])^2 = 1 + 0 = 1 \]

3. Third Moment

\[ E[X^3] = 0 \]

(odd moment of a symmetric distribution around 0)

4. Fourth Moment

The fourth moment of a standard normal is:

\[ E[X^4] = 3 \]

This comes from the identity for the fourth moment of a normal variable:

\[ E[X^4] = 3\sigma^4 \quad \text{when } X \sim N(0, \sigma^2) \]

Final Results

\[ \boxed{ E[X] = 0,\quad E[X^2] = 1,\quad E[X^3] = 0,\quad E[X^4] = 3 } \]

Reference