(MATHS)Probability Question

1. A biased coin is tossed repeatedly, with p the probability of heads on any single toss.Find the probability that there is a run of r heads in a row before there is a run of stails, where r and s are positive integers.
2. Let X have the Uniform distribution on [0; 1]. For what function g does Y = g(X)have the exponential distribution with parameter ? Explain.
3 A hen lays N eggs, where N has the Poisson distribution with parameter . Each egg hatches with probability p independently of the other eggs. Let K be the number of chicks. What is the conditional expectation of the initial number N of eggs, given that we observe k chicks?

4. Mrs. Jones has made a pie for her two sons. Eating more than a half of it will give indigestion to anyone. While she is away having tea with a neighbor, the older son helps himself a piece of the pie. Then the younger son comes and has a piece of what is left by his brother. Assume that the size of each of the two pieces eaten by Mrs. Jones' sons is random and uniformly distributed over what is currently available. What is the expected size of the remaining piece given that neither son gets indigestion?

5.Suppose that the outcome X of a certain chance mechanism depends on a parameter P according to P(X = 1) = P and P(X = 0) = 1 ���� P, where P 2 [0; 1]. Suppose that P is chosen at random, uniformly distributed over the unit interval [0; 1], and then, that two independent outcomes X1 and X2 are observed. What is the unconditional covariance between X1 and X2? Comment on the sign of this value and explain why this is not in contradiction with the assumed independence between X1 and X2.