Problem 1

Find an example of two discrete random variables X and Y (on the same sample space) such that X and Y have the same distribution (i.e., same density and same CDF), but the event X = Y never occurs.

Problem 2

Consider the following game: first we press a button, and a computer gives us a random variable \(X\sim U[0,10]\). If X is less then 4 we roll a fair die, otherwise we roll a die which gives the number i proportional to i2. (that means P(i)=c*i2 for some constant c). Let Y be the number rolled. Find the distribution function of Y.

Problem 3

Consider the following random variable X: we roll a fair die. If the number i comes up \(X\sim U[0,i]\). Find the distribution function of X. Draw the graph.

Problem 4

Consider a density of the following shapes:

so f increases quadraticaly from 0 to some a and then decreases linearly to 0 at 1. Find an expression for the distribution function F(x;a). Find F(0.6; 0.2). (Note that the quadratic function has a vertex at 0).