A number is chosen at random from the set [math]S = \{-1,0,1\}[/math]. Let [math]X[/math] be the number chosen. Find the expected value, variance, and standard deviation of [math]X[/math].

A random variable [math]X[/math] has the distribution

Find the expected value, variance, and standard deviation of [math]X[/math].

You place a 1-dollar bet on the number 17 at Las Vegas, and your friend places a 1-dollar bet on black (see exercise and exercise). Let [math]X[/math] be your winnings and [math]Y[/math] be her winnings. Compare [math]E(X)[/math], [math]E(Y)[/math], and [math]V(X)[/math], [math]V(Y)[/math]. What do these computations tell you about the nature of your winnings if you and your friend make a sequence of bets, with you betting each time on a number and your friend betting on a color?

[math]X[/math] is a random variable with [math]E(X) = 100[/math] and [math]V(X) = 15[/math]. Find

• [math]E(X^2)[/math].
• [math]E(3X + 10)[/math].
• [math]E(-X)[/math].
• [math]V(-X)[/math].
• [math]D(-X)[/math].

In a certain manufacturing process, the (Fahrenheit) temperature never varies by more than [math]2^\circ[/math] from [math]62^\circ[/math]. The temperature is, in fact, a random variable [math]F[/math] with distribution

• Find [math]E(F)[/math] and [math]V(F)[/math].
• Define [math]T = F - 62[/math]. Find [math]E(T)[/math] and [math]V(T)[/math], and compare these answers with those in part (a).
• It is decided to report the temperature readings on a Celsius scale, that is, [math]C = (5/9)(F - 32)[/math]. What is the expected value and variance for the readings now?

Write a computer program to calculate the mean and variance of a distribution which you specify as data. Use the program to compare the variances for the following densities, both having expected value 0:

A coin is tossed three times. Let [math]X[/math] be the number of heads that turn up. Find [math]V(X)[/math] and [math]D(X)[/math].

A random sample of 2400 people are asked if they favor a government proposal to develop new nuclear power plants. If 40 percent of the people in the country are in favor of this proposal, find the expected value and the standard deviation for the number [math]S_{2400}[/math] of people in the sample who favored the proposal.

A die is loaded so that the probability of a face coming up is proportional to the number on that face. The die is rolled with outcome [math]X[/math]. Find [math]V(X)[/math] and [math]D(X)[/math].

Prove the following facts about the standard deviation.

• [math]D(X + c) = D(X)[/math].
• [math]D(cX) = |c|D(X)[/math].