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Chapter 2: Problem 1

Let \(X, Y, Z\) have joint pdf \(f(x, y, z)=2(x+y+z) / 3,0

Short Answer

Expert verified
Before sharing the actual numbers, this problem requires multiple integrations to find marginal PDFs, testing a factorization to check independence, and computing expected values and probabilities. Finally, finding conditional distributions and expectations involves dividing joint distributions.

Step by step solution

01

Find the Marginal PDFs

To find the marginal probability density functions (PDFs) for \(X\), \(Y\), and \(Z\), integrate the joint PDF \(f(x, y, z)\) over the remaining variables. For instance, the marginal PDF of \(X\) (denoted by \(f_{X}(x)\)) is found by integrating \(f(x, y, z)\) over \(y\) and \(z\) on their respective intervals [0, 1]: \(f_{X}(x)= \int_{0}^{1} \int_{0}^{1} f(x, y, z) dy dz\). Similarly, find \(f_{Y}(y)\) and \(f_{Z}(z)\) by integrating over \(x\) and \(z\), and \(x\) and \(y\), respectively.
02

Compute Probabilities

Compute the probabilities \(P\left(0<X<\frac{1}{2}, 0<Y<\frac{1}{2}, 0<Z<\frac{1}{2}\right)\) and \(P\left(0<X<\frac{1}{2}\right)=P(0<Y<\frac{1}{2})=P\left(0<Z<\frac{1}{2}\right)\) by integrating the respective marginal PDFs over the specified ranges.
03

Test for Independence

Check if \(X\), \(Y\), and \(Z\) are independent. This is the case if and only if the joint PDF factorizes into a product of the marginal PDFs, i.e., \(f(x, y, z) = f_{X}(x) f_{Y}(y) f_{Z}(z)\). If this holds true for all \(x, y, z\) in the support of the random variables, then \(X\), \(Y\), and \(Z\) are independent.
04

Compute Expected Values

Compute the expected value of \(E\left(X^{2} Y Z+3 X Y^{4} Z^{2}\right)\) by integrating this expression times \(f(x,y,z)\) over all values of \(x, y, z\).
05

Find the CDFs

Determine the cumulative distribution functions (CDFs) of \(X\), \(Y\), and \(Z\) by integrating their respective PDFs from their lower limits to a variable upper limit.
06

Find Conditional Distributions and Expectations

Determine the conditional distributions of \(X\) and \(Y\), given \(Z=z\), by dividing the joint distribution of \(X\), \(Y\), and \(Z\) by the marginal distribution of \(Z\). Evaluate the expected value \(E(X+Y \mid z)\) by integrating \(x+y\) times the conditional PDF over all \(x\) and \(y\). Similarly, find the conditional distribution of \(X\), given \(Y=y\) and \(Z=z\), by dividing the joint distribution of \(X\), \(Y\), and \(Z\) by the joint distribution of \(Y\) and \(Z\). Compute the expected value \(E(X \mid y, z)\) by integrating \(x\) times the conditional PDF over all \(x\).

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Most popular questions from this chapter

Let us choose at random a point from the interval \((0,1)\) and let the random variable \(X_{1}\) be equal to the number which corresponds to that point. Then choose a point at random from the interval \(\left(0, x_{1}\right)\), where \(x_{1}\) is the experimental value of \(X_{1}\); and let the random variable \(X_{2}\) be equal to the number which corresponds to this point. (a) Make assumptions about the marginal pdf \(f_{1}\left(x_{1}\right)\) and the conditional pdf \(f_{2 \mid 1}\left(x_{2} \mid x_{1}\right)\) (b) Compute \(P\left(X_{1}+X_{2} \geq 1\right)\). (c) Find the conditional mean \(E\left(X_{1} \mid x_{2}\right)\).

Let \(f_{1 \mid 2}\left(x_{1} \mid x_{2}\right)=c_{1} x_{1} / x_{2}^{2}, 0

Let \(X\) and \(Y\) be random variables with the space consisting of the four points \((0,0),(1,1),(1,0),(1,-1)\). Assign positive probabilities to these four points so that the correlation coefficient is equal to zero. Are \(X\) and \(Y\) independent?

If the independent variables \(X_{1}\) and \(X_{2}\) have means \(\mu_{1}, \mu_{2}\) and variances \(\sigma_{1}^{2}, \sigma_{2}^{2}\), respectively, show that the mean and variance of the product \(Y=X_{1} X_{2}\) are \(\mu_{1} \mu_{2}\) and \(\sigma_{1}^{2} \sigma_{2}^{2}+\mu_{1}^{2} \sigma_{2}^{2}+\mu_{2}^{2} \sigma_{1}^{2}\), respectively.

Let \(X_{1}, X_{2}, X_{3}, X_{4}\) be four iid random variables having the same pdf \(f(x)=\) \(2 x, 0
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