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).

Binomial random variables

Let random variable X be a binomial random variable with parameters n and p, i.e., let X be the number of successful outcomes we get by performing experiment that has success probability p for n times independently.

For $1\leq i\leq n$, define a random variable X_i to be 1 if the i-th experiment is successful and 0 otherwise.

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

2. Find $\mathrm {E} [X_{i}\cdot X_{j}]$.

3. From the definition, we have that $X=\sum _{i=1}^{n}X_{i}$. Rewrite $X^{2}$ in terms of $X_{i}^{2}$ and $X_{i}\cdot X_{j}$.

4. Use the expansion in (3) to find $\mathrm {E} [X^{2}]$ and $\mathrm {Var} (X)$.