Problem 2

How many time do we need to toss to estimate the fairness of a coin?

Checking whether a coin is fair

Problem 1

The expectation times of sampling to get a number twice from a discrete uniform distribution $U(1,n)$.

$P(X = x)$ denotes the probability that $S = \{s_1, ..., s_x\}$ are distinct numbers, and $s_{x+1} \in S$.

Solution

Expectation:

where

Variance:

I don’t find a good way to calculate it yet.

Problem 0

Sample $m$ numbers $\{x_1, \dots, x_m\}$ without replacement from $\{1, \dots, n+m\}$. Let $X = x_1 + \dots + x_m$. Find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.