[SOLVED] Math154 homework 4- independence p0

MATH154
Homework 4
Independence
b) What is the probability that Ana has a meeting during a time that Bobhas a meeting?
c) What is the probability that Bob has a meeting during a time whenAna has a meeting?
d) Is the event that both have a meeting at the same time independent ofthe event that both have no meeting at the same time?

Figure 1. Palindromes Ana and Bob meet to discuss the design of new dice. (AI generated picture)
Probability Theory
Problem 4.2: True or False? (Please give justifications).
1) If A,B are independent, then A,Bc are independent.
2) If P[B] > 0, and A,B are independent, then P[A|B] = P[A].
3) If A,B are independent and B,C are independent then A,C are.
4) If A,B,C are independent, then A + B is independent of C.
5) If A,B,C are independent, then A ∩ B is independent of C.
6) If A,B,C are independent then A ∪ B is independent of C. 7) Two disjoint sets A,B are independent if and only if P[A] = 0 or P[B] = 0.
8) ∅ is independent of any other set.
9) Ω is independent of any other set.
10) If A is independent to itself, then P[A] = 0 or P[A] = 1.
Problem 4.3: If (Ω,A,P) has a P trivial σ-algebra, you might think that A is the trivial σ-algebra. This is not the case as you verify here with an example:
Verify that the σ algebra of cocountable or countable sets in Ω = [0,1] is
P-trivial, if P = λ is the probability Lebesgue measure on [0,1]
Problem 4.4: In all of this problem, all random variables are bounded L∞.
a) Verify that if X,Y are independent and n,m are positive integers, then Xn,Y m are independent.
b) Verify that X · Y = ⟨X,Y ⟩ = E[XY ] defines an inner product on L2.

Define |X| = p⟨X,X⟩. Check Cauchy-Schwarz |⟨X,Y ⟩| ≤ |X||Y |.
c) We have seen that if X,Y are independent L2 random variables, then E[XY ] = E[X]E[Y ]. Can you reverse this? Does the condition E[XY ] = E[X]E[Y ] imply that X,Y are independent?
d) What about asking that E[XnY m] = E[Xn]E[Y m] for all n,m > 0?
Does this imply that X,Y are independent?
Problem 4.5: a) Verify that the moment generating function of the Cauchy distribution does not exist.
b) Compute the characteristic function ϕX(t) of a Cauchy distributed random variable.
c) Compute the characteristic function of the Gaussian distribution with
−x2/√π. probability
density function f(x) = e
d) Find a probability space and a random variable X such that ϕX(t) = cos(t).