Let (N, L, P) be a probability space and H EL with P(H) > 0. For any arbitrary subset, A E L, and define P(AN H) P(H) PH(A) = P(A|H) Then show that (H, LH, PH) is a probability space.

Let S = {X1, X2, X3, X4, X5} be a sample space with probability function P(A). Let E₁ = {x}. Assume that P(E2) = P(E4) = 2P(E1) and P(E1) = P(E3) = P(E5). Find a. P(E) for all i b. P({X1, X3}) C. P({X2, X3, X4})

Let (Ω, F, P) be a probability space. Show that the following holds for all n ≥ 1 and allevents A1, A2, . . . , An ∈ F:

Let F(x) be the cumulative distribution function (cdf) of a random variable X. Which of the following is/are true? F(x) is a non-increasing function. X is a discrete random variable. If a, b, and X are integers, and a F(b). Y is a continuous random variable. If a < b, then F(a) + Pr(a < Y < b) = F(b). O O O O

Consider a probability space (2, F, P) and consider a random variable, X Let Fx (x) be the distribution function of X Prove that Fx (t) - limno Fx (t-n-¹)=P{ X=t}.

Please do part 3

Contracts for two construction jobs are randomly assigned to one of or more of three firms, A, B, and C. Let Y1 denote the number of contracts assigned to firm A and Y2 the number of contracts assigned to firm B. Recall that each firm can receive 0, 1, or 2. a) Find the joint probability function for Y1 and Y2. b) Find F(1,0).

Let (Ω, F, P) be a probability space and let A, B, C be events with P(C) > 0. We call A andB (conditional) independent given C if and only ifP(A ∩ B|C) = P(A|C)P(B|C).(a) Show that conditional independence can be understood as independence with respect to a suitably chosen probability measure P˜ on (Ω, F).(b) There are two coins in a box. One coin is a fair coin,the other coin is a fake coin with a head on both sides. Weblindly take a coin from the box and toss it. Then we throw itremaining coin. We watch the eventsA = 'the first coin lands heads';B = 'the second coin lands heads';C = 'the coin that is taken first is the fair coin'.Determine whether the following statements and explain why they are true or false:1. A and B are independent.2. A and B are given independently C.Provide both intuitive reasoning and formal proof.

Please do the second part

A drawer holds purple socks and yellow socks. If n socks are taken out of the drawer at random, the probability that all are yellow is 1/2. What is the smallest possible number of socks in the drawer (as a function of n)?