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Solution: Let the joint probability density function of X
Chapter , Problem 3(choose chapter or problem)
Let the joint probability density function of X and Y be bivariate normal. For what values of \(\alpha\) is the variance of \(\alpha X+Y\) minimum?
Questions & Answers
QUESTION:
Let the joint probability density function of X and Y be bivariate normal. For what values of \(\alpha\) is the variance of \(\alpha X+Y\) minimum?
ANSWER:Step 1 of 2
Given that the joint probability density function of X and Y is bivariate normal.
To find the value of \(\alpha\) for which the variance of \(\alpha X+Y\) is minimum.
Let us consider
\(\begin{array}{l}X\sim N\left(\mu_X,\ \sigma_X^2\right)\\ Y\sim N\left(\mu_Y,\ \sigma_Y^2\right)\end{array}\)
The correlation coefficient is \(\rho\). Then,
It is known that
\(Var(aX+bY)=a^2Var(X)+b^2Var(Y)+2abCov(X,\ Y)\)