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Chapter 2: Problem 3

Let \(f(x, y)=2,0

Short Answer

Expert verified
The conditional mean of X given Y is \( (1+x) / 2, 0 < x < 1 \), the conditional mean of Y given X is \( y / 2, 0 < y < 1 \), and the correlation coefficient of X and Y is \( \rho = 1 / 2 \).

Step by step solution

01

Compute the Marginal Distributions

Compute the marginal pdfs \(f_X(x)\) and \(f_Y(y)\). This is done by means of integration. The marginal pdf of X is obtained by integrating the joint pdf over Y and similarly for the marginal pdf of Y by integrating over X: \(\int_{x}^{1} f(x, y) dy\) for \(f_X(x)\) and \(\int_{0}^{y} f(x, y) dx\) for \(f_Y(y)\)
02

Calculate the Conditional Means

The conditional means \(E(X|Y=y)\) and \(E(Y|X=x)\) are computed. The conditional expectation \(E(X|Y=y)\) is given by \(\int x \cdot f_{X|Y}(x|y) dx\) and for \(E(Y|X=x)\) it is given by \(\int y \cdot f_{Y|X}(y|x) dy\). Substituting the marginal distributions calculated in step 1, these integrals are solved to obtain the conditional means.
03

Calculation of the Correlation Coefficient

The correlation coefficient is given by the formula: \(\rho = \frac{Cov(X,Y)}{\sigma_X \sigma_Y} \). The covariance can be found by: \(Cov(X,Y) = E[XY] - E[X]E[Y]\) and the standard deviations of X and Y can be computed using their variance formulas. Once these are calculated, they can be plugged into the correlation formula to find the correlation coefficient.

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Most popular questions from this chapter

Let \(f(x, y)=e^{-x-y}, 0

Let \(X_{1}, X_{2}\) have the joint pdf \(f\left(x_{1}, x_{2}\right)=1 / \pi, 0

Let \(X_{1}, X_{2}, X_{3}\) be iid with common pdf \(f(x)=e^{-x}, x>0,0\) elsewhere. Find the joint pdf of \(Y_{1}=X_{1} / X_{2}, Y_{2}=X_{3} /\left(X_{1}+X_{2}\right)\), and \(Y_{3}=X_{1}+X_{2}\). Are \(Y_{1}, Y_{2}, Y_{3}\) mutually independent?

Let \(X_{1}, X_{2}\), and \(X_{3}\) be three random variables with means, variances, and correlation coefficients, denoted by \(\mu_{1}, \mu_{2}, \mu_{3} ; \sigma_{1}^{2}, \sigma_{2}^{2}, \sigma_{3}^{2}\); and \(\rho_{12}, \rho_{13}, \rho_{23}\), respectively. For constants \(b_{2}\) and \(b_{3}\), suppose \(E\left(X_{1}-\mu_{1} \mid x_{2}, x_{3}\right)=b_{2}\left(x_{2}-\mu_{2}\right)+b_{3}\left(x_{3}-\mu_{3}\right)\) Determine \(b_{2}\) and \(b_{3}\) in terms of the variances and the correlation coefficients.

Let \(X_{1}, X_{2}, X_{3}\), and \(X_{4}\) be four independent random variables, each with pdf \(f(x)=3(1-x)^{2}, 0y)=P\left(X_{i}>y, i=1, \ldots, 4\right)\).
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