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Chapter 7: Q.23 (page 360)

Show that Zis a standard normal random variable and if Yis defined by Y=a+bZ+cZ2, then

蚁(Y,Z)=bb2+2c2

Short Answer

Expert verified

We prove thatY=a+bZ+cZ2

Step by step solution

01

Given information

Given in the question that,Y=a+bZ+cZ2.

02

Explanation

Let Zis a standard normal random variable N(0,1), and Y=a+bZ+cZ2

Now, it can be shown蚁(Y,Z)

蚁(Y,Z)=Cov(Y,Z)Var(Y)Var(Z)

=bVar(Z)+cEZ3-cEZ2E(Z)

E(Z)=0,Var(Z)=1, because Zare standard normal random variable.

EZ3=0order moments are always zero,

Cov(Y,Z)=bVar(Z)=b

Var(Y)=b2Var(Z)+c2VarZ2

=b2+c2EZ4-c2EZ22

=b2+3c2-1c2

=b2+2c2

蚁(Y,Z)=bb2+2c21=bb2+2c2

Hence, we obtain:

蚁(Y,Z)=bb2+2c2

03

Final answer

We prove thatY=a+bZ+cZ2

and

蚁(Y,Z)=bb2+2c2

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