Thread: I need help with my Probability Course assignment

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 five 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.

Read this. Rule No 6.

You are supposed to do your homework by yourself!

This thread is destined to be closed.

Originally Posted by harish21

Read this. Rule No 6.

You are supposed to do your homework by yourself!

This thread is destined to be closed.

Rule #8 would also be relevant if the questions were not part of a graded assignment.