Problem 1.

There are 2 Random samples from [0, 1] uniform distribution. What is the average distance between them.

Problem 2.

Let $U$ be a continuous uniform $[0,1]$ random variable. What is the probability that the decimal expansion of $U$ contains no fives?

Problem 3.

$X$ is the $N(0,1)$ random variable. What is the distribution density function of $Y = X^2$?

Problem 4.

An unknown proportion $p$ of the electorate will vote Labour. It is desired to find $p$ without an error not exceeding $0.005$. How large should the sample be?

Problem 5.

One person rolls 100 fair dies and calculates the sum of all the outcomes. Denote the resulting random variable as $D$. Another one tosses 600 fair coins, and calculates the total number of heads on them. Denote the resulting random variable as $C$. Estimate the probability $D > C$.