Pick a point [math]x[/math] at random (with uniform density) in the interval [math][0,1][/math]. Find the probability that [math]x \gt 1/2[/math], given that

• [math]x \gt 1/4[/math].
• [math]x \lt 3/4[/math].
• [math]|x - 1/2| \lt 1/4[/math].
• [math]x^2 - x + 2/9 \lt 0[/math].

A radioactive material emits [math]\alpha[/math]-particles at a rate described by the density function

Find the probability that a particle is emitted in the first 10 seconds, given that

• no particle is emitted in the first second.
• no particle is emitted in the first 5 seconds.
• a particle is emitted in the first 3 seconds.
• a particle is emitted in the first 20 seconds.

The Acme Super light bulb is known to have a useful life described by the density function

where time [math]t[/math] is measured in hours.

• Find the failure rate of this bulb (see exercise).
• Find the reliability of this bulb after 20 hours.
• Given that it lasts 20 hours, find the probability that the bulb lasts another 20 hours.
• Find the probability that the bulb burns out in the forty-first hour, given that it lasts 40 hours.

Suppose you toss a dart at a circular target of radius 10 inches. Given that the dart lands in the upper half of the target, find the probability that

• it lands in the right half of the target.
• its distance from the center is less than 5 inches.
• its distance from the center is greater than 5 inches.
• it lands within 5 inches of the point [math](0,5)[/math].

Suppose you choose two numbers [math]x[/math] and [math]y[/math], independently at random from the interval [math][0,1][/math]. Given that their sum lies in the interval [math][0,1][/math], find the probability that

• [math]|x - y| \lt 1[/math].
• [math]xy \lt 1/2[/math].
• [math]\max\{x,y\} \lt 1/2[/math].
• [math]x^2 + y^2 \lt 1/4[/math].
• [math]x \gt y[/math].

Find the conditional density functions for the following experiments.

• A number [math]x[/math] is chosen at random in the interval [math][0,1][/math], given that [math]x \gt 1/4[/math].
• A number [math]t[/math] is chosen at random in the interval [math][0,\infty)[/math] with exponential density [math]e^{-t}[/math], given that [math]1 \lt t \lt 10[/math].
• A dart is thrown at a circular target of radius 10 inches, given that it falls in the upper half of the target.
• Two numbers [math]x[/math] and [math]y[/math] are chosen at random in the interval [math][0,1][/math], given that [math]x \gt y[/math].

Let [math]x[/math] and [math]y[/math] be chosen at random from the interval [math][0,1][/math]. Show that the events [math]x \gt 1/3[/math] and [math]y \gt 2/3[/math] are independent events.

Let [math]x[/math] and [math]y[/math] be chosen at random from the interval [math][0,1][/math]. Which pairs of the following events are independent?

• [math]x \gt 1/3[/math].
• [math]y \gt 2/3[/math].
• [math]x \gt y[/math].
• [math]x + y \lt 1[/math].

Suppose that [math]X[/math] and [math]Y[/math] are continuous random variables with

density functions [math]f_X(x)[/math] and [math]f_Y(y)[/math], respectively. Let [math]f(x, y)[/math] denote the joint density function of [math](X, Y)[/math]. Show that

and

In Exercise you proved the following: If you take a stick of unit length and break it into three pieces, choosing the breaks at random (i.e., choosing two real numbers independently and uniformly from [0, 1]), then the probability that the three pieces form a triangle is 1/4. Consider now a similar experiment: First break the stick at random, then break the longer piece at random. Show that the two experiments are actually quite different, as follows:

• Write a program which simulates both cases for a run of 1000 trials, prints out the proportion of successes for each run, and repeats this process ten times. (Call a trial a success if the three pieces do form a triangle.) Have your program pick [math](x,y)[/math] at random in the unit square, and in each case use [math]x[/math] and [math]y[/math] to find the two breaks. For each experiment, have it plot [math](x,y)[/math] if [math](x,y)[/math] gives a success.
• Show that in the second experiment the theoretical probability of success is actually [math]2\log 2 - 1[/math].