Question

Contracts for two construction jobs are randomly assigned to one or more of three firms A, B, and C. Let Y1 denote the number of contracts assigned to firm A and Y2 the number of contracts assigned to firm B. Recall that each firm can receive 0, 1 or 2 contracts.

(a) Find the joint probability function for Y1 and Y2.

(b) Find the marginal probability of Y1 and Y2.

(c) Are Y1 and Y2 independent? Why?

(d) Find E(Y1 − Y2).

(e) Find Cov(Y1, Y2)

Answer 1

There are three firms. Job is randomly assigned to one of the firms randomly i.e

Probability of a job assigned to one of the given firms =1/3

Probability of assigning two jobs to any of the firms =(1/3)(1/3)=1/9

(Example : Probability of assigning first job to Firm A and second job firm B =1/3x1/3=1/9 ; Similarlry ,

Probability of assigning first job to Firm C and second job firm B =1/3 x 1/3 =1/9...so on)

(a) Joint Probability function of Y1, Y2

(b)

Marginal Probability of Y1

Marginal of Y2 =

(c)

Therefore, Y1 and Y2 are independent

(d)

E(Y1-Y2) = E(Y1)-E(Y2)

E(Y1-Y2) = E(Y1)-E(Y2) =1-1=0

E(Y1-Y2) = 0

(e)

As Y1, Y2 are independent,

E(Y1Y2) = E(Y1)E(Y2)

Cov(Y1Y2) = E(Y1Y2) - E(Y1)E(Y2) = 0

Cov (Y1Y2) = 0

(In can be observed that,

E(Y1Y2) =1

E(Y1)=1

E(Y2)=1

Cov(Y1Y2) = E(Y1Y2) - E(Y1)E(Y2) =1-1x1 =1-1=0

Cov(Y1,Y2) =0