91影视

We already know that one of our objectives for this session is to determine the sample mean's probability distribution when a random sample is obtained from a population with normally distributed data. Let's get right to the punch line, shall we? So, let's start with a probability distribution for a linear combination of independent normal random variables.

Given: \({\rm{X}}\)and \({\rm{Y}}\) are standard normal random variables that are independent.

There is no association between two random variables that are independent:

\({\mathop{\rm Corr}\nolimits} (X,Y) = 0\)

A variable has a one-to-one correlation with itself:

\(\begin{array}{l}{\mathop{\rm Corr}\nolimits} (X,X) = 1\\{\mathop{\rm Corr}\nolimits} (Y,Y) = 1\end{array}\)

Property correlation:

\({\rm{Corr(E,cA + dB) = cCorr(E,A) + dCorr(E,A)}}{\rm{.}}\)

Using the prior properties, we can determine the correlation between \({\rm{X}}\)and \({\rm{Y}}\):

\(\begin{aligned} Corr(X,U) &= Corr(X,0 {\rm{.6X + 0}}{\rm{.8Y)}}\\ &= 0 {\rm{.6Corr(X,X) + 0}}{\rm{.8Corr(X,Y)}}\\ &= 0 {\rm{.6(1) + 0}}{\rm{.8(0) = 0}}{\rm{.6}}\end{aligned}\)

(b)

Assume that \(U = aX + bY\).

Using the properties described in part (a), we can determine the correlation between \({\rm{X}}\)and \({\rm{Y}}\):

\(\begin{aligned} Corr(X,U) &= Corr(X,aX + bY)\\ &= aCorr(X,X) + bCorr(X,Y) \\ &= a(1) + b(0) = a\end{aligned}\)

If we want the correlation to be equal to \(\rho \),we must set the value of \(a\)to \(\rho \) (the value of \(b\)can be set at random, for example, \(b = 1\)):

\({\rm{a = \rho }}\)

\({\rm{b = 1}}\)

Then we obtain:

\({\rm{U = \rho X + Y}}\)

\({\rm{Cov(X,U) = 0}}{\rm{.6}}\)Therefore, the result is :

\({\rm{U = \rho X + Y}}\)