1) If E[X] = , what is E[X ]? use an example to guide.Try to prove using properties of Expectation.

2) If E[X] = 5 and E[Y] = 12, find using properties

(a) E[3X + 4Y]

(b) E[2 + 5Y + X]

(c) E[4 + Y]

3) Two fair coins are tossed. Determine Var(X) when X is the number of heads that appear.

(a) Use the definition of the variance.

(b) Use the fact that the variance of the sum of independent random variables is equal to the sum of the variances.

4) Individuals who have two sickle cell genes will develop the disease called sickle cell anemia, while individuals having none or one sickle cell gene will not be harmed. If two people, both of whom have one sickle cell gene, have a child, then that child will receive two sickle cell genes with probability 1/4. Suppose that both members of each of three different couples have exactly one sickle cell gene. If each of these couples has a child, find the probability that

(a) None of the children receives two sickle cell genes.

(b) Exactly one of the children receives two sickle cell genes.

(c) Exactly two of the children receive two sickle cell genes.

(d) All three children receive two sickle cell genes.

5) Four fair dice are to be rolled. Find the probability that

(a) 6 appears at least once.

(b) 6 appears exactly once.

(c) 6 appears at least twice.