Think of one example of a random variable which is non-degenerate for which all the odd moments are identically zero. Give the probability mass function of the random variable and state a quantity it could represent.
At first I thought of rolling a die since it's non-degenerate, but I don't believe its odd moments are 0. Is anyone able to come up with an example and briefly explain it?
1. Continuous Random Variable
A standard gaussian, $X\sim N(0;1)$ works.
$$\mathbb{E}[X^{2n+1}]=0$$
$\forall n \in \mathbb{N}$
The proof is quite easy expanding its MGF in Taylor series and derivating
It can represent the measurement error when measuring the length of the following stick
2. Discrete Random Variable
$Y$ is a random variable taking the values $Y=\pm1$ with probability $\mathbb{P}[Y=-1]=\mathbb{P}[Y=1]=\frac{1}{2}$
$$\mathbb{E}[Y^{2n+1}]=\frac{1}{2}[(-1)^{2n+1}+1^{2n+1}]=0$$
$\forall n \in \mathbb{N}$
$Y$ represents the following function
$$Y=2X-1$$
Where $X\sim B\Big(\frac{1}{2}\Big)$, a Bernoulli rv with parameter 0.5
It can represent the random gain when playing "toss a fair coin game" winning $\$1$ if H and loosing $\$1$ if T