Write a program to simulate the random variables of Exercise and Exercise and plot a bar graph of the results. Compare the resulting empirical density with the density found in Exercise and Exercise.

A number [math]U[/math] is chosen at random in the interval [math][0,1][/math]. Find the probability that

• [math]R = U^2 \lt 1/4[/math].
• [math]S = U(1 - U) \lt 1/4[/math].
• [math]T = U/(1 - U) \lt 1/4[/math].

Find the cumulative distribution function [math]F[/math] and the density function [math]f[/math] for each of the random variables [math]R[/math], [math]S[/math], and [math]T[/math] in Exercise.

A point [math]P[/math] in the unit square has coordinates [math]X[/math] and [math]Y[/math] chosen at random in the interval [math][0,1][/math]. Let [math]D[/math] be the distance from [math]P[/math] to the nearest edge of the square, and [math]E[/math] the distance to the nearest corner. What is the probability that

• [math]D \lt 1/4[/math]?
• [math]E \lt 1/4[/math]?

In Exercise find the cumulative distribution [math]F[/math] and density [math]f[/math] for the random variable [math]D[/math].

Let [math]X[/math] be a random variable with density function

• What is the value of [math]c[/math]?
• What is the cumulative distribution function [math]F_X[/math] for [math]X[/math]?
• What is the probability that [math]X \lt 1/4[/math]?

Let [math]X[/math] be a random variable with cumulative distribution function

• What is the density function [math]f_X[/math] for [math]X[/math]?
• What is the probability that [math]X \lt 1/4[/math]?

Let [math]X[/math] be a random variable with cumulative distribution function [math]F_X[/math], and let [math]Y = X + b[/math], [math]Z = aX[/math], and [math]W = aX + b[/math], where [math]a[/math] and [math]b[/math] are any constants. Find the cumulative distribution functions [math]F_Y[/math], [math]F_Z[/math], and [math]F_W[/math]. Hint: The cases [math]a \gt 0[/math], [math]a = 0[/math], and [math]a \lt 0[/math] require different arguments.

Let [math]X[/math] be a random variable with density function [math]f_X[/math], and let [math]Y = X + b[/math], [math]Z = aX[/math], and [math]W = aX + b[/math], where [math]a \ne 0[/math]. Find the density functions [math]f_Y[/math], [math]f_Z[/math], and [math]f_W[/math]. (See Exercise.)

Let [math]X[/math] be a random variable uniformly distributed over [math][c,d][/math], and let [math]Y = aX + b[/math]. For what choice of [math]a[/math] and [math]b[/math] is [math]Y[/math] uniformly distributed over [math][0,1][/math]?