Solution Found!
Let Y1 and Y2 be jointly distributed random variables with
Chapter 5, Problem 167SE(choose chapter or problem)
Let \(Y_{1} \text { and } Y_{2}\) be jointly distributed random variables with finite variances.
a Show that \(\left[E\left(Y_{1} Y_{2}\right)\right]^{2} \leq E\left(Y_{1}^{2}\right) E\left(Y_{2}^{2}\right)\) [Hint: Observe that \(E\left[\left(t Y_{1}-Y_{2}\right)^{2}\right] \geq 0\) for any real number 𝑡 or, equivalently,
\(t^{2} E\left(Y_{1}^{2}\right)-2 t E\left(Y_{1} Y_{2}\right)+E\left(Y_{2}^{2}\right) \geq 0.\)
This is a quadratic expression of the form \(A t^{2}+B t+C\); and because it is nonnegative, we must have \(B^{2}-4 A C \leq 0\). The preceding inequality follows directly.]
b Let \(\rho\) denote the correlation coefficient of \(Y_{1} \text { and } Y_{2}\). Using the inequality in part (a), show that \(\rho^{2} \leq 1\).
Equation Transcription:
Text Transcription:
Y_1 and Y_2
[E(Y_1Y_2)]^2E(Y_1^2)E(Y_2^2)
E[(tY_1-Y_2)^2]>/=0
t^2E(Y_1^2)-2tE(Y_1Y_2)+E(Y_2^2)>/=0.
At^2+Bt+C
B^2-4AC</=0
rho
Y1 and Y2
rho^2</=1
Questions & Answers
QUESTION:
Let \(Y_{1} \text { and } Y_{2}\) be jointly distributed random variables with finite variances.
a Show that \(\left[E\left(Y_{1} Y_{2}\right)\right]^{2} \leq E\left(Y_{1}^{2}\right) E\left(Y_{2}^{2}\right)\) [Hint: Observe that \(E\left[\left(t Y_{1}-Y_{2}\right)^{2}\right] \geq 0\) for any real number 𝑡 or, equivalently,
\(t^{2} E\left(Y_{1}^{2}\right)-2 t E\left(Y_{1} Y_{2}\right)+E\left(Y_{2}^{2}\right) \geq 0.\)
This is a quadratic expression of the form \(A t^{2}+B t+C\); and because it is nonnegative, we must have \(B^{2}-4 A C \leq 0\). The preceding inequality follows directly.]
b Let \(\rho\) denote the correlation coefficient of \(Y_{1} \text { and } Y_{2}\). Using the inequality in part (a), show that \(\rho^{2} \leq 1\).
Equation Transcription:
Text Transcription:
Y_1 and Y_2
[E(Y_1Y_2)]^2E(Y_1^2)E(Y_2^2)
E[(tY_1-Y_2)^2]>/=0
t^2E(Y_1^2)-2tE(Y_1Y_2)+E(Y_2^2)>/=0.
At^2+Bt+C
B^2-4AC</=0
rho
Y1 and Y2
rho^2</=1
ANSWER:
Solution :
Step 1 of 2:
Let and
be the jointly distributed random variable with finite variances.
Our goal is:
a). We need to show that
b). We need to show that .
The quadratic form of interest is
The integral of the non-negative quantity is .
So we must have that
Then the two roots of this quadratic must either be imaginary or equal.
Now in terms of the discriminant, we have that
or
So
Therefore