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Chapter 2: Problem 8

Let \(X\) and \(Y\) have the joint pdf \(f(x, y)=3 x, 0

Short Answer

Expert verified
To find if \(X\) and \(Y\) are independent, we would test if the joint pdf equals to the product of the marginal pdfs. If not, we can then find the conditional expectation \(E(X \mid y)\) by integrating \(x \cdot f_{X|Y}(x|y) \, dx\) over all possible values of \(X\)

Step by step solution

01

Determine Independence

Calculate the marginal pdfs of \(X\) and \(Y\) from the joint pdf \(f(x, y)=3x\), for \(0<y<x<1\) and then check if the joint pdf equals the product of the marginal pdfs. The marginal pdf of \(Y\), denoted as \(f_Y(y)\), can be obtained by integrating the joint pdf over all possible values of \(X\). Similarly, the marginal pdf of \(X\), \(f_X(x)\), can be obtained by integrating the joint pdf over all possible values of \(Y\). Subsequently, if \(f(x, y) = f_X(x)f_Y(y)\), for all \(x\) and \(y\), then the variables are independent.
02

Calculating E(X|y)

Assuming that the variables are not independent, we need to calculate \(E(X \mid y)\), the expected value of \(X\) given \(Y = y\). This is accomplished by integrating \(x \cdot f_{X|Y}(x|y) \, dx\) over all possible values of \(X\). Where \(f_{X|Y}(x|y)\) is the conditional pdf of \(X\) given \(Y = y\), and can be found by dividing the joint pdf \(f(x, y)\) by the marginal pdf of \(Y\), \(f_Y(y)\).
03

Stip 3: Solve and Interpret

Finally, the solutions derived from the above steps are to be interpreted. If the joint pdf is equal to the product of the marginal pdfs for all values of \(x\) and \(y\), \(X\) and \(Y\) are independent. If not, the calculated \(E(X \mid y)\) would provide the expected value of \(X\) given \(Y = y\)

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Most popular questions from this chapter

Let \(X\) and \(Y\) have the joint \(\mathrm{pmf} p(x, y)=\frac{1}{7},(0,0),(1,0),(0,1),(1,1),(2,1)\), \((1,2),(2,2)\), zero elsewhere. Find the correlation coefficient \(\rho .\)

Let \(f(x)\) and \(F(x)\) denote, respectively, the pdf and the cdf of the random variable \(X\). The conditional pdf of \(X\), given \(X>x_{0}, x_{0}\) a fixed number, is defined by \(f\left(x \mid X>x_{0}\right)=f(x) /\left[1-F\left(x_{0}\right)\right], x_{0}x_{0}\right)\) is a pdf. (b) Let \(f(x)=e^{-x}, 02 \mid X>1)\).

Let \(X\) and \(Y\) have the joint pmf described as follows: $$\begin{array}{c|cccccc}(x, y) & (1,1) & (1,2) & (1,3) & (2,1) & (2,2) & (2,3) \\ \hline p(x, y) & \frac{2}{15} & \frac{4}{15} & \frac{3}{15} & \frac{1}{15} & \frac{1}{15} & \frac{4}{15} \end{array}$$ and \(p(x, y)\) is equal to zero elsewhere. (a) Find the means \(\mu_{1}\) and \(\mu_{2}\), the variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\), and the correlation coefficient \(\rho\). (b) Compute \(E(Y \mid X=1), E\left(Y \mid X=2\right.\) ), and the line \(\mu_{2}+\rho\left(\sigma_{2} / \sigma_{1}\right)\left(x-\mu_{1}\right) .\) Do the points \([k, E(Y \mid X=k)], k=1,2\), lie on this line?

Let \(X_{1}, X_{2}\) be two random variables with joint \(\mathrm{pmf} p\left(x_{1}, x_{2}\right)=(1 / 2)^{x_{1}+x_{2}}\), for \(1 \leq x_{i}<\infty, i=1,2\), where \(x_{1}\) and \(x_{2}\) are integers, zero elsewhere. Determine the joint mgf of \(X_{1}, X_{2}\). Show that \(M\left(t_{1}, t_{2}\right)=M\left(t_{1}, 0\right) M\left(0, t_{2}\right)\).

Let 13 cards be talsen, at random and without replacement, from an ordinary deck of playing cards. If \(X\) is the number of spades in these 13 cards, find the pmf of \(X\). If, in addition, \(Y\) is the number of hearts in these 13 cards, find the probability \(P(X=2, Y=5) .\) What is the joint pmf of \(X\) and \(Y ?\)
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