Pretoria University Random Variables Probability Questions

5. [8 points] Consider three events: E₁, E2, and F. Assume that P(F) > 0, and thatP(E₁ E₂F) = P(E₁F)P(E2|F). Is it necessarily always the case that P(E₁ E₂) =
P(E₁)P(E₂)? Prove or find a counterexample.
6. [10 points] Let the random variable S and T be independent and identically distributed
as follows:
P(S = -1) = P(T = -1) = P(S = 1) = P(T = 1) = 0.5.
Define the random variable R as the product R = ST.
(a) [5 points] Is R independent of S and T?
(b) [5 points] Is R independent of S + T?
7. [10 points] Bob has an urn which contains a black balls and b green balls. Bob picks
one ball at a time, without replacing the ball into the urn, until he has c black balls where
c≤a. What is the probability that Bob needs n ball draws to reach c black balls?
8. [10 points] Carl decides to sell his old laptop on eBay, and sequentially receives bids
from potential buyers. The minimum price that he will accept to sell his laptop for is
£500. Let {X, n ≥ 0} denote the sequence of independent and identically distributed
bids that Carl receives, and assume that each X,, has the following probability density
function
fx(x) = (1/400)e-/400 for x ≥ 0.
Let N denote the number of bids that Carl obtains before selling his laptop i.e., Carl sells
his laptop to the Nth bid.
(a) [5 points] What is E[N]?
(b) [5 points] What is E[XN]?
9. [10 points] Consider the trajectory of a particle on an (x, y) plane. Let (X, Yn) be the
position of the particle at the nth step, n ≥ 0. The particle is initially at position (0,0)
i.e., Xo = 0 and Y₁ = 0. Let its sequential movements, i.e., between steps n and n + 1,
be defined as follows:
P(Xn+1 =i-1, Yn+1 =jXn = i, Y₁ = j) = P(Xn+1=i+1, Yn+1 =jXni, Yn = j) = 1/4,
P(Xn+1 = i, Yn+1 = j-1|X₂ = i, Y₁ = j) = P(Xn+1 = i, Yn+1 =j+1|X₂ = i, Y₁ = j) = 1/4.
(a) [5 points] Are X₁, and Y₁, independent?
(b) [5 points] Calculate Cov(X₁,Y₁).