Suppose the prices in Y1 Y2 Y3 YA of objects A B C and D are

Suppose the prices (in Y1, Y2, Y3, YA of objects A, B, C, and D are jointly normally distributed as 3 2 3 3 2 5 5 4 and 3 59 5 Y N (A, 3), where Y 3 4 5 6 Answer the following questions a. Suppose a person wants to buy three of product A, four of product B, and one of product C Find the probability that the person will spend more than $30. b. Find the moment generating function of Yi c. Find the joint moment generating function of (yi,Y3). d. Find the correlation coefficient between Ya and Y

Solution

a) We require find the probability that ,

P((3*Y1 + 4*Y2 + Y3)>30)

    Now, we know that if X,Y,Z follows a multivariate normal distribution then any linear combination will follow a univariate normal distribution,

So take Z=3Y₁+4Y₂+Y₃

And E(Z)=21, Var(Z)=222

Therefore, Z ~ N(21,222)

Now the required probability is,
P(Z>30))

=P((Z-21)/sqrt(222) >((30-21)/sqrt(222)))

=P(T>0.604), where T~ N(0,1)

=0.27426

b) Here Y1 follows a univariate normal distribution so the MGF is

M(t)=exp(t+1.5*t^2)

c) Here, Y1 and Y3 are jointly follows bivariate normal distribution,

So the mgf of Y1 and Y3 is,

M(t1,t2)=exp(t1 + 6*t2 +(3*t1*t1 + 9*t2*t2 + 2*3*t1*t2)/2)

d) Correlation(Y3,Y4)= cov(Y3,Y4)/sqrt(var(Y3)*Var(Y4))

Here, cov(Y3,Y4)=3

Var(Y3)=9, Var(Y4)=6

So the answer is 0.40824829