Problem 2 (40 points): (A D) 10pts A discrete RV, X, can take values: -2,-1,0, 1

Question

Problem 2 (40 points): (A D) 10pts A discrete RV, X, can take values: -2,-1,0, 1, and 2 with equal probabilities. Define a RV, Y-X a) Fill up the joint probability table, P(X,Y) below X/Y 0 2 P(Y) 0 PO b) Find EX) E(Y). VoX) and V(Y) c) Are X and Y independent? d Find Cov(X,Y) and Cor(X,Y) (Covariance and Correlation). Briefly discuss the results. Problem 3 (35 points): (A C) 10pts D) 5pts A popular pizza shop has two web-pages to receive online orders. The first web-page has Poisson orders with 2 arrivals per minute. The 2d web-page receives orders on the average 1 minute time in-between that can be modelled as Exponential distribution. Obviously both web-pages are independent from each other. (a) What is the joint probability distribution function for the order waiting times? (b) What is the probability that no orders will be received in a 2-minute period? In a 30 seconds period? (c) What is the probability that both web-sites will receive two orders betweent and 4th minutes after they are officially online for business? (d) To answer the questions (b) and (c) do you need to calculate the joint probabilities? Why?

Explanation / Answer

Question 2:

a) The joint probability distribution for X and Y here are defined as:

Note that there are fixed Y values for a given X here, therefore rest of the probabilities in the above table are 0.

a) E(X) = 0.2*(-2-1+0+1+2) = 0

Therefore E(X) = 0

E(Y) = 0.4*(1 + 4) = 2

Therefore E(Y) = 2

E(X2) = 0.2*(4 + 1 + 0 + 1 +4) = 2
E(Y2) = 0.4*(1 + 16) = 6.8

Therefore, Var(X) = E(X2) - [ E(X)]2 = 2 - 0 = 2
Var(Y) = E(Y2) - [ E(Y)]2 = 6.8 - 22 = 2.8

Therefore Var(X) = 2 and Var(Y) = 2.8

c) P(X = 0, Y = 0) = 0.2

P(X = 0)P(Y= 0) = 0.2*0.2 = 0.04 not equal to P(X = 0, Y = 0)

Therefore X and Y are not independent here.

d) E(XY) = E(X3) = 0.2(-8 -1 + 0 + 1 + 8) = 0

Therefore, Cov(X, Y) = E(X)E(Y) = 0

Therefore Cov(X, Y) = 0

and therefore Cor(X, Y) = 0

X = -2 X = -1 X = 0 X = 1 X = 2 P(Y) Y = 0 0 0 0.2 0 0 0.2 Y = 1 0 0.2 0 0.2 0 0.4 Y = 4 0.2 0 0 0 0.2 0.4 P(X) 0.2 0.2 0.2 0.2 0.2

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