Probstat/week4 practice 2

This is part of probstat.

Useful properties

1. For a random variable X and constants a and c, prove that E[aX+c]=aE[X] + c.

2. Let X be a random variable and $\mu ={\mathrm {E} }[X]$, prove that ${\mathrm {E} }[X-\mu ]=0$.

Dinner dish experiment

Recall the dinner experiment: n people are having a dinner at the table with a rotating turntable. Each person orders one different dish and gets her/his order exactly in front of her/him. They decide to randomly rotate the turntable so that each one of them will get a random dish. Let random variable Y be the number of people who get their own dish after the random rotation.

1. Find ${\mathrm {E} }[Y^{2}]$.

We define the variance Var(Y) of a random variable Y whose expectation ${\mathrm {E} }[Y]=\mu$ to be ${\mathrm {E} }[(Y-\mu )^{2}]$.

2. Find Var(Y).