Question

Let ?1 and ?2 be independent normal random variables with ?1~ ?(µ, ? 2 ) and ?2~ ?(µ, ? 2 ). Let ?1 = ?1 and ?2 = 2?2 − ?1.

a) Find ??1,?2 (?1,?2).

b) What are the means, variances, and correlation coefficient for ?1 and ?2?

c) Find ?2(?2).

Answer 1

SOLUTION :

given that

ler X1 and X2 are two independent f normal variables

,   

The transformed variables are given as follows:

and   

then

To obtain the results based on Y₁ and Y₂, we first need to find the probability function of two variables Y₁and Y₂.

Now since ,    Therefore,   

Further, implies that

Mean of X_1,                  and             And Variance of X₁,

and we know that,            

                                              

Thus, we can conclude .

Also, =>

Mean of X_2,                  and             And Variance of X₂,

and we know that,            

                                              

Thus, we can conclude .

We know that the sum and difference of finite number of normal variables is also normal variate. Such as,   and are normal random variable, then

Thus, this implies, if

        and           .

Then

That is,

Now based on these outcomes, we have to obtain the following results:

(a).

The random variables Y₁ and Y₂ are independent normal random variables and for independent random variables, we have,

We know that the mgf for is given by

Thus the mgf of and will be

                and                   

Hence,

                         

                         

                         

(b). Here

Thus Mean of Y₁                  and             And Variance of Y₁

And

Thus Mean of Y₂                  and             And Variance of Y₂

The correlation coefficient between two random variables Y₁ and Y₂ is given by,

But the two variables Y₁ and Y₂ are independent therefore the covariance Cov (Y₁, Y₂ ) between the two variables is zero and thus the correlation coefficient r is also zero

(c).

The random variable Y₂ follows normal distribution such as . Thus we write probability function for Y₂ as follows:

             

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