Alright I've got a probability test coming up and our professor gaves us a work sheet with 30+ problems to review and these are

some I'm having trouble with and could use alittle direction. Thanks to anyone who even reads my post. Well here they are:

Problem 1.

A fair die having two faces colored blue, two red, and two green, is thrown repeatedly. Find the probability that not all colors occur in the rst n throws.

Now suppose that N is the random variable that takes the value n if n is the rst time that all three colors have occurred. Find the expected value of N.

I've made myself get pretty confused on this one and ended up just starting to right out the possibilities and count, something on the lines of when n=0,1,2 p=0, n=3 p=7/9, n=4 p=5/9, n=5 p=31/81, n=6 p=7/27, but i'm not even sure about those anymore I'm so over the place. Any ideas would be much appreciated.

Problem 4. If A1,A2,...Am are independent and P(Ai) = p for i = 1,2,...m, find the probability that an even number of the A's occur.

Problem 9. The random variable X has a continuous distribution function FX(x). Suppose that Z =FX(X). What is the distribution of Z?

Problem 10. Now suppose that X be a random variable which is uniformly distributed on the interval (-1, 1). Compute the distribution function of Y = max[0,X]?

Problem 11. The bilateral exponential distribution has density function

f(x) = (c/2)e^(-c|x|) , x all Reals

where c > 0 is a constant. Compute its mean and its variance.

Problem 12. The random variables X and Y each take values -1 and 1, and P(X = 1) = a, P(Y = 1) = b. Suppose Z = XY and that X, Y and Z are pairwise independent? What are the values of a and b? Are

{X,Y,Z} independent?

Problem 15. Each time you fiip a certain coin, heads appears with probability p. Suppose that you fiip the coin a random number N of times, where N has the Poisson distribution with parameter and is independent of the outcomes of the fiips. Find the distributions of the numbers of X and Y of the heads

and tails, respectively. Are X and Y independent?

Again thank you for trying to help out. I've honestly been sitting for hours trying to find out how to do each of these. Yea probability!