Problem 2 (10 points). There are two bank tellers, A and B. Let X represent the

Question

Problem 2 (10 points). There are two bank tellers, A and B. Let X represent the number of people waiting in line for teller A, and Y represent the number of people waiting in line for teller B. X and Y follow the joint pmf shown below: 4 0 0.05 0.06 0.05 0.02 0.01 10.07 0.15 0.04 0.03 0.01 20.04 0.05 0.10 0.04 0.05 3 0.01 0.04 0.06 0.07 0.05 (a) Calculate the marginal pmfs of X and Y (b) Determine Cov(X, Y) and pxy (c) What is the probability that the number of people waiting for each teller differs by 2? (d) What is the probability that an equal number of people are waiting for each teller?

Explanation / Answer

here joint distribution of X and Y:

a)

marginal pmf of X is given in f(x) column against values of X:

marginal pmf of Y is given in f(y) column against values of Y:

b)

E(XY) =xyP(x,y) =3.21

hence Cov(X,Y) =E(XY)-E(X)*E(Y) =0.4820

correlation coefficient pxy =Cov(X,Y) /(Var(X)*Var(Y))1/2 =0.3698

c) P(|X-Y|=2) =P(X=0;Y=2)+P(X=1;Y=3)+P(X=2,Y=0)+P(X=3;Y=1)+P(X=4;Y=2)

=0.04+0.04+0.05+0.03+0.05=0.2100

d)P(equal number) =P(X=0;Y=0)+P(X=1;Y=1)+P(X=2;Y=2)+P(X=3;Y=3)=0.3700

x y 0 1 2 3 4 Total 0 0.0500 0.0600 0.0500 0.0200 0.0100 0.1900 1 0.0700 0.1500 0.0400 0.0300 0.0100 0.3000 2 0.0400 0.0500 0.1000 0.0400 0.0500 0.2800 3 0.0100 0.0400 0.0600 0.0700 0.0500 0.2300 Total 0.1700 0.3000 0.2500 0.1600 0.1200 1.0000

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