Be $X ∼ N$ $(µ, σ^2)$ a normally distributed random variable on a probability space $(Ω, \cal F, \cal P)$ Can someone help me to calculate the following expected values:
i) $\mathbb E[X^{2k+1}]$ for $k ∈ \mathbb N$, if $ µ = 0 $.
ii)$\mathbb E[X^{2k}]$ for $k ∈ \mathbb N$, if $ µ = 0 $.
I know I need to calculate: $\mathbb E[X^{2k+1}] = \int_{-\infty}^{\infty} x^{2k+1} \cdot f(x)dx$ and
$\mathbb E[X^{2k}] = \int_{-\infty}^{\infty} x^{2k} \cdot f(x)dx$
with: $ f(x)\ =\ \frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-(x)^2}{2\sigma^2}} $
But I can't necessarily calculate the integrals. We didn't have the gamma function either. Can you help me? Can I assume $ \sigma=1 $ since $ µ = 0 $? Thanks in advance
$Y=\frac {X-\mu} {\sigma} \sim N(0,1)$. Write $X$ as $\sigma Y+\mu$ and expand $(\sigma Y+\mu)^{n}$ using Binomial Theorem.