Choose a number [math]U[/math] from the unit interval [math][0,1][/math] with uniform distribution. Find the cumulative distribution and density for the random variables

• [math]Y = U + 2[/math].
• [math]Y = U^3[/math].

Choose a number [math]U[/math] from the interval [math][0,1][/math] with uniform distribution. Find the cumulative distribution and density for the random variables

• [math]Y = 1/(U + 1)[/math].
• [math]Y = \log(U + 1)[/math].

Use Corollary to derive the expression for the random variable given in Equation. Hint: The random variables [math]1 - rnd[/math] and [math]rnd[/math] are identically distributed.

Suppose we know a random variable [math]Y[/math] as a function of the uniform random variable [math]U[/math]: [math]Y = \phi(U)[/math], and suppose we have calculated the cumulative distribution function [math]F_Y(y)[/math] and thence the density [math]f_Y(y)[/math]. How can we check whether our answer is correct? An easy simulation provides the answer: Make a bar graph of [math]Y = \phi(\mbox{$rnd$})[/math] and compare the result with the graph of [math]f_Y(y)[/math]. These graphs should look similar. Check your answers to Exercise and Exercise by this method.

Choose a number [math]U[/math] from the interval [math][0,1][/math] with uniform distribution. Find the cumulative distribution and density for the random variables

• [math]Y = |U - 1/2|[/math].
• [math]Y = (U - 1/2)^2[/math].

Check your results for Exercise by simulation as described in Exercise.

Explain how you can generate a random variable whose cumulative distribution function is

Write a program to generate a sample of 1000 random outcomes each of which is chosen from the distribution given in Exercise. Plot a bar graph of your results and compare this empirical density with the density for the cumulative distribution given in Exercise.

Let [math]U[/math], [math]V[/math] be random numbers chosen independently from the interval [math][0,1][/math] with uniform distribution. Find the cumulative distribution and density of each of the variables

• [math]Y = U + V[/math].
• [math]Y = |U - V|[/math].

Let [math]U[/math], [math]V[/math] be random numbers chosen independently from the interval [math][0,1][/math]. Find the cumulative distribution and density for the random variables

• [math]Y = \max(U,V)[/math].
• [math]Y = \min(U,V)[/math].