Chapter 2: Problem 6
Let \(X\) and \(Y\) have the joint pdf \(f(x, y)=1,-x
Chapter 2: Problem 6
Let \(X\) and \(Y\) have the joint pdf \(f(x, y)=1,-x
All the tools & learning materials you need for study success - in one app.
Get started for freeLet \(f(x)\) 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}
Let the random variables \(X_{1}\) and \(X_{2}\) have the joint pmf described as follows: $$\begin{array}{c|cccccc}\left(x_{1}, x_{2}\right) & (0,0) & (0,1) & (0,2) & (1,0) & (1,1) & (1,2) \\ \hline f\left(x_{1}, x_{2}\right) & \frac{2}{12} & \frac{3}{12} & \frac{2}{12} & \frac{2}{12} & \frac{2}{12} & \frac{1}{12} \end{array}$$ and \(f\left(x_{1}, x_{2}\right)\) 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 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 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 \(A_{1}=\\{(x, y): x \leq 2, y \leq 4\\}, A_{2}=\\{(x, y): x \leq 2, y \leq
1\\}, A_{3}=\)
\(\\{(x, y): x \leq 0, y \leq 4\\}\), and \(A_{4}=\\{(x, y): x \leq 0 y \leq
1\\}\) be subsets of the
space \(\mathcal{A}\) of two random variables \(X\) and \(Y\), which is the entire
two-dimensional plane. If \(P\left(A_{1}\right)=\frac{7}{8},
P\left(A_{2}\right)=\frac{4}{8}, P\left(A_{3}\right)=\frac{3}{8}\), and
\(P\left(A_{4}\right)=\frac{2}{8}\), find \(P\left(A_{5}\right)\), where
\(A_{5}=\\{(x, y): 0
What do you think about this solution?
We value your feedback to improve our textbook solutions.