How to find the fourth-order moment of the sample mean

Question

r.v. $X_1, \ldots, X_n \sim \text{i.i.d. } N(\theta, 1)$.

target

$$ \mathbb E \big[(\bar{X} - \theta)^4\big] \overset?= 3/n^2 $$ where $\bar{X}$ means sample mean.

Please give me some ideas for easy calculations.

Answer 1

r.v. $Y \sim N(\mu, \sigma^2), \mu \in \mathbb R, \sigma > 0.$

$$ \mathbb E[(Y-\mu)^m] = (2k-1)\cdots 5\cdot 3\cdot1\cdot \sigma^{2k} \mathrm{\ \ \ \ if\ } m = 2k, k=1, 2, \ldots $$

Now $\bar{X} \sim N(\theta, 1/n).$