Exercise 1: [Conditional probabilities]
Consider two events A and B of a probability space. Prove the following:
(a) Conditioned inclusion-exclusion P(A ∪ B | C) = P(A | C) + P(B | C) - P(A ∩ B | C).
(b) Bayes formula:
P(B | A) = P(A | B) P(B)/P(A|B) P(B) + P(A | Bc) P(Bc)
(c) If B is an event with full probability, i.e. P(B) = 1, then
i. P(A ∩ B) = P(A).
ii. P(A | B) = P(A).
Exercise 2: [Almost partitions]
Given a probability space, consider a countable (or finite) family of events B1, B2, . . . such that
(A) they are disjoint: Bi ∩ Bj = ∅ for i ≠ j, and
(B) they "almost" cover S: P(∪i Bi) = 1.
(a) Prove that the "divide-and-conquer" formula also holds, namely
P(A) = ∑iP(A | Bi) P(Bi)
for every A ∈ Σ.
(b) Let ΣB the smallest σ-algebra containing the Bi's.
i. Describe all the events in ΣB.
ii. Determine the general form of ΣB-measurable random variables.
Exercise 3: [Sum of independent binomials]
Let X1, X2, . . . , XN be independent random variables where Xi is a binomial of parameters ni and p. Find the law of Z = X1 + X2 + · · · + Xn.
Exercise 4: [Conditioned Poisson]
Let X and Y be Poisson random variables with respective rates λ1 and λ2, and let Z = X + Y.
(a) Find E(X | Z).
(b) Verify that E[E(X | Z)] = E(X).
Exercise 5: [Composed randomness]
Let X be a random variable distributed with an exponential law with a random rate λ. This rate is itself distributed uniformly in the interval [1, 2]. Find.
(a) P(X > x).
(b) E(X).
Exercise 6: [Alternative definition of Markovianness]
Prove that if (Xn)n≥0 is a Markov process if and only if
P(Xn+k = xn+k | Xn = xn, Xn-1 = xn-1, . . . , X0 = x0) = P(Xn+k = xn+k | Xn = xn) for all n, k ≥ 0.
Exercise 7:
Consider a Markov chain with state space {1, 2, 3, 4} and transition matrix
(a) Show that Pn4 4 = (1/3)n.
(b) Show that the state "4" is transient.
(c) Let T = inf{n > 0 : Xn ≠ 4} be the time it takes the process to exit "4" (for ever). Compute E(T | X0 = 4). [Hint: you may want to use that for a discrete random variable Z, E[Z] = ∑k≥0 P(Z > k).]
(d) Let T3 = inf{n > 0 : Xn = 3} be the absorption time at state "3". Compute P(T = T3 | X0 = 4), that is the probability that the process exist "4" only to be absorbed by "3".
Exercise 8: [Markovian fishing]
At a certain fishing spot two consecutive rainy day are twice as likely to be followed by another rainy day than by a sunny day, while two consecutive sunny day are five times more likely to be followed by another sunny day. It one of two consecutive days is rainy and the other sunny, they are equally like to be sunny or rainy. A fisherman observes that the number of catches per day is Poisson distributed with mean 4 in rainy days and mean 1 in sunny days. He visits the spot on Saturday and Sunday, which are both sunny day.
(a) Find the probability that Tuesday is a sunny day.
(b) Find the mean number of fish expected for Tuesday.