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Chapter 2: Problem 1

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

Short Answer

Expert verified
The conditional mean of \(X_2\), given \(X_1 = x_1\), is \(\frac{2}{3}\), and the conditional variance of \(X_2\), given \(X_1 = x_1\), is \(\frac{1}{18}\).

Step by step solution

01

Compute Conditional PDF

Firstly, let's find the marginal pdf of \(X_{1}\), \(f_{X_1}(x_{1})\):\[f_{X_1}(x_{1}) = \int^{1}_{0}f(x_{1},x_{2})dx_{2}=\int^{1}_{0}(x_{1} + x_{2})dx_{2}=x_{1} + \frac{1}{2}.\]Next, using \(f_{X_2|X_1}(x_{2}=v|x_{1}=x_1) = \frac{f_{X_1, X_2}(x_{1}, v)}{f_{X_1}(x_{1})}\) the conditional pdf of \(X_2\) given \(X_1 = x_1\) is:\[ f_{X_2|X_1}(x_{2}=v|x_{1}=x_{1}) = \frac{x_{1} + v}{x_{1}+\frac{1}{2}}, 0<= v <= 1. \]
02

Compute Conditional Mean

The conditional mean of \(X_2\) given \(X_1 = x_1\) is:\[ E[X_2|X_1 = x_1] = \int^{1}_{0} v \cdot f_{X_2\|X_1}(v|x_{1})dv = \int^{1}_{0}v \cdot \frac{x_{1} + v}{x_{1}+\frac{1}{2}} dv = \frac{2}{3}. \]
03

Compute Conditional Variance

The conditional variance of \(X_2\) given \(X_1 = x_1\) is:\[ Var[X_2|X_1 = x_1] = E[X_2^2|X_1 = x_1] - \left(E[X_2|X_1 = x_1]\right)^2 = \int^{1}_{0} v^2 \cdot f_{X_2|X_1}(v|x_{1})dv - \left(\frac{2}{3}\right)^2. \]This simplifies to:\[ Var[X_2|X_1 = x_1] = \int^{1}_{0} v^2 \cdot \frac{x_{1} + v}{x_{1}+\frac{1}{2}} dv - \frac{4}{9}= \frac{1}{18}. \]

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Most popular questions from this chapter

Let the random variables \(X_{1}\) and \(X_{2}\) have the joint pmf described as follows: $$ \begin{array}{c|cccccc} \left(x_{1}, x_{2}\right) & (0,0) & (0,1) & (0,2) & (1,0) & (1,1) & (1,2) \\ \hline p\left(x_{1}, x_{2}\right) & \frac{2}{12} & \frac{3}{12} & \frac{2}{12} & \frac{2}{12} & \frac{2}{12} & \frac{1}{12} \end{array} $$ and \(p\left(x_{1}, x_{2}\right)\) is equal to zero elsewhere. (a) Write these probabilities in a rectangular array as in Example 2.1.3, recording each marginal pdf in the "margins." (b) What is \(P\left(X_{1}+X_{2}=1\right)\) ?

Let \(X\) and \(Y\) have the parameters \(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}\), and \(\rho\). Show that the correlation coefficient of \(X\) and \(\left[Y-\rho\left(\sigma_{2} / \sigma_{1}\right) X\right]\) is zero.

Use the formula \((2.2 .2)\) to find the pdf of \(Y_{1}=X_{1}+X_{2}\), where \(X_{1}\) and \(X_{2}\) have the joint pdf \(f_{X_{1}, X_{2}}\left(x_{1}, x_{2}\right)=2 e^{-\left(x_{1}+x_{2}\right)}, 0

Determine the mean and variance of the sample mean \(\bar{X}=5^{-1} \sum_{i=1}^{5} X_{i}\), where \(X_{1}, \ldots, X_{5}\) is a random sample from a distribution having pdf \(f(x)=\) \(4 x^{3}, 0

If \(X\) has the pdf of \(f(x)=\frac{1}{4},-1
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