Probstat/week6 practice 1

This is part of probstat.

In this practice, we will try a few experiments that shows distributions of random variables with varying variances.

Variances and distribution

In each of the following experiments, you should find the expectation and the variance of the specified random variable. Then perform at least 10,000 experiments to generate the histogram of the distribution of the random variable.

Set 1: uniform distribution

1.1 We pick randomly an integer from the set {0,1,2,...,10}. Let X be the number that we picked. Find E[X], Var(X), and perform random experiments to plot X histogram.

Hint: for question 1.2 - 1.5, finding Var(X) 'indirectly' using the result from 1.1 might be an easier approach.

1.2 We pick randomly an integer from the set {0,1,2,...,10} for 2 times. Let Y be the sum of the 2 numbers that we picked. Let X = Y/2. Find E[X], Var(X), and perform random experiments to plot X histogram.

1.3 We pick randomly an integer from the set {0,1,2,...,10} for 3 times. Let Y be the sum of the 3 numbers that we picked. Let X = Y/3. Find E[X], Var(X), and perform random experiments to plot X histogram.

1.4 We pick randomly an integer from the set {0,1,2,...,10} for 5 times. Let Y be the sum of the 5 numbers that we picked. Let X = Y/5. Find E[X], Var(X), and perform random experiments to plot X histogram.

1.5 We pick randomly an integer from the set {0,1,2,...,10} for 10 times. Let Y be the sum of the 10 numbers that we picked. Let X = Y/10. Find E[X], Var(X), and perform random experiments to plot X histogram.