I have got a few questions for my assignment. I can't seem to solve 9 of these. I have no other way of getting help for these. I know these will come up for my exams. Please help me solve them.

1. Suppose S is uncountable. Show that it is impossible that P({s}) > 0 for every s that belongs to S.

2. Suppose we choose a positive integer at random, according to some unknown probability distribution. Suppose we know that P({1, 2, 3, 4, 5}) = 0.3 and that P({4, 5, 6}) = 0.4 and that P({1}) = 0.1. What are the largest and smallest possible values of P({2})?

3. Suppose there are C people, each of whose birthdays (month and day only) are equally likely to fall on any of the 365 days of a normal year. Suppose C greater than or equal to 2.

a. What is the probability that all C people have the same exact birthday?

b. What is the probability that some pair of the C people have the same exact birthday?

4. Let A and B be events of positive probability. Prove that P(A|B) > P(A) if and only if P(B|A) > P(B).

5. Let P be some probability measure on sample space S = [0, 1].

a. Prove that we must have lim(n-->1) P((0, 1/n)) = 0.

b. Show by example that we might have lim(n-->1) P([0, 1/n)) > 0.

6. Let X be a random variable.

a. Is it necessarily true that there is some real number c such that (X + c) is greater than or equal to 0?

b. Suppose the sample space S is finite. Then is it necessarily true that there is some real number c such

that (X + c) is greater than or equal to 0?

7. Suppose Alice flips three fair coins, and let X be the number of heads showing. Suppose Barbara flips five fair coins, and let Y be the number of heads showing. Let Z = X - Y . Compute P(Z = k) for every real number k.

8. Consider the function given by f(x) = (e^|x|)/2 for -infinity < x < +infinity. Prove that f is a density function.

9. Let X be a random variable, with cumulative distribution function Fx. Prove that P(X = a) = 0 if and only if the function Fx is continuous at a.