1. Consider the density function

f(x) = {k√x,

0 < x < 1,

elsewhere .

a)   Evaluate k.

b)   Find F(x) and use it to evaluate P(0.3 < X < 0.6) .

2. From a box containing 4 dimes and 2 nickels, 3 coins are selected at random without replacement. Find the probability distribution for the total T of the 3 coins. Express the probability distribution graphically   as a probability histogram.

3. The total time, measured in units of 100 hours, that a teenager runs her hair dryer over a period of one year is a continuous random variable X that has the density function

f(x) = {2 − x,

0 < x < 1,

1 ≤ x < 2,

elsewhere.

Use Theorem 4.6 to evaluate the mean of the random variable Y = 60X2  + 39X , where Y is equal to the number of kilowatt hours expended annually.

4. Determine the values of c so that the following functions represent joint probability distributions of the random variables X and Y.

a) f(x, y) = cxy, for x = 1,2,3; y = 1,2,3;

b) f(x, y) = c|x − y|, for x = −2,0,2; y = −2,3.

5. Let X denote the diameter of an armored electric cable and Y denote the diameter of the ceramic mold that makes the cable. Both X and Y are scaled so that they range between 0 and 1. Suppose that X and Y have the joint density

f(x) = { ,

Find P (X + Y > ).

0 < x < y < 1,

elsewhere.

6. A chemical system that results from a chemical reaction has two important components among others in a blend. The joint distribution describing the proportions X1  and X2  of these two components is given by

f(X1, X2) = {

0 < X1  < X2  < 1,

elsewhere.

a)   Give the marginal distribution of X1 .

b)   Give the marginal distribution of X2 .

c)   What is the probability that component proportions produce the results X1< 0.2 and X2> 0.5?

d)   Give the conditional distribution fX1|X2 (X1 |X2).

7. A large industrial firm purchases several new word processors at the end of each year, the exact     number depending on the frequency of repairs in the previous year. Suppose that the number of word processors, X, purchased each year has the following probability distribution:

If the cost of the desired model is $1200 per unit and at the end of the year a refund of 50X2  dollars will be issued, how much can this firm expect to spend on new word processors during this year?

8. Suppose that the probabilities are 0.4, 0.3, 0.2, and 0.1, respectively, that 0, 1, 2, or 3 power failures  will strike a certain subdivision in any given year. Find the mean and variance of the random variable X representing the number of power failures striking this sub-division.

9. Suppose that a grocery store purchases 5 cartons of skim milk at the wholesale price of $1.20 per carton and retails the milk at $1.65 per carton. After the expiration date, the unsold milk is removed from the shelf and the grocer receives a credit from the distributor equal to three-fourths of the wholesale price. If the probability distribution of the random variable X, the number of cartons that are sold from this lot,  is

find the expected profit.

10. Let X represent the number that occurs when a red die is tossed and Y the number that occurs when a green die is tossed. Find

a)   E(X+Y);

b)   E(X-Y);

c)   E(XY).

11. Let X be a random variable with the following probability distribution: