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Chapter 2: Problem 3

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

Short Answer

Expert verified
The conditional means are \((1+x)/2\) for \(0<x<1\) and \(y/2\) for \(0<y<1\). The correlation coefficient \(\rho\) for \(X\) and \(Y\) is \(\frac{1}{2}\).

Step by step solution

01

Find Conditional Mean of X

First, find the marginal density of \(Y\) by integrating the joint density function \(f(x, y)\) over \(x\) from 0 to \(y\):\[f_Y(y) = \int_0^y 2 dx = 2y, 0<y<1\]Then, the conditional mean of \(X\) given \(Y = y\) is calculated by:\[\mathbb{E}(X|Y = y) = \int_0^y x \frac{f(x, y)}{f_Y(y)} dx = \int_0^y \frac{x}{y} dx = \frac{1}{2}(1+x), 0<x<1\]
02

Find Conditional Mean of Y

Similar to the above step, the marginal density of \(X\) is obtained by integrating \(f(x, y)\) over \(y\), from \(x\) to 1:\[f_X(x) = \int_x^1 2 dy = 2(1-x), 0<x<1\]Afterward, we compute the conditional mean of \(Y\) given \(X = x\), which is:\[\mathbb{E}(Y|X = x) = \int_x^1 y \frac{f(x, y)}{f_X(x)} dy = \int_x^1 \frac{y}{1-x} dy = \frac{y}{2}, 0<y<1\]
03

Calculate the Correlation Coefficient

Finally, calculate the correlation coefficient (\(\rho\)). To do this, find the means of \(X\) and \(Y\) first: \[\mathbb{E}(X) = \int_0^1 x f_X(x) dx, \mathbb{E}(Y) = \int_0^1 y f_Y(y) dy\]Then, calculate the expectation of the product of \(X\) and \(Y\):\[\mathbb{E}(XY) = \int \int xy f(x, y) dx dy\]The covariance is given by:\[\text{Cov}(X, Y) = \mathbb{E}(XY) - \mathbb{E}(X)\mathbb{E}(Y)\]Lastly, find the standard deviations of \(X\) and \(Y\) and apply these to find the correlation coefficient:\[\rho = \frac{\text{Cov}(X, Y)}{\text{sd}(X)\cdot \text{sd}(Y)}\]Follow these instructions and you will find \(\rho = \frac{1}{2}\).

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Most popular questions from this chapter

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

Let \(\mu\) and \(\sigma^{2}\) denote the mean and variance of the random variable \(X\). Let \(Y=c+b X\), where \(b\) and \(c\) are real constants. Show that the mean and variance of \(Y\) are, respectively, \(c+b \mu\) and \(b^{2} \sigma^{2}\).

Let \(X_{1}, X_{2}, X_{3}\) be iid with common mgf \(M(t)=\left((3 / 4)+(1 / 4) e^{t}\right)^{2}\), for all \(t \in R\) (a) Determine the probabilities, \(P\left(X_{1}=k\right), k=0,1,2\). (b) Find the mgf of \(Y=X_{1}+X_{2}+X_{3}\) and then determine the probabilities, \(P(Y=k), k=0,1,2, \ldots, 6\)

Let the joint pdf of \(X\) and \(Y\) be given by $$ f(x, y)=\left\\{\begin{array}{ll} \frac{2}{(1+x+y)^{3}} & 0

Let \(X_{1}\) and \(X_{2}\) have the joint pmf described by the following table: \begin{tabular}{c|cccccc} \(\left(x_{1}, x_{2}\right)\) & \((0,0)\) & \((0,1)\) & \((0,2)\) & \((1,1)\) & \((1,2)\) & \((2,2)\) \\ \hline\(p\left(x_{1}, x_{2}\right)\) & \(\frac{1}{12}\) & \(\frac{2}{12}\) & \(\frac{1}{12}\) & \(\frac{3}{12}\) & \(\frac{4}{12}\) & \(\frac{1}{12}\) \end{tabular} Find \(p_{1}\left(x_{1}\right), p_{2}\left(x_{2}\right), \mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}\), and \(\rho .\)
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