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3.69. Given the values of the joint probability distribu- tion of X and Y shown in the table -1 yo 1 7 0 1 8 X 1 1 2 1 0 find (a) the marginal distribution of X; (b) the marginal distribution of Y; (c) the conditional distribution of X given Y = -1.

3.69. Given the values of the joint probability distribu- tion of X and Y shown in the table -1 yo 1 7 0 1 8 X 1 1 2 1 0 find (a) the marginal distribution of X; (b) the marginal distribution of Y; (c) the conditional distribution of X given Y = -1.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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### Problem 3.69 Given the values of the joint probability distribution of \(X\) and \(Y\) shown in the table: \[ \begin{array}{c|cc} & x = -1 & x = 1 \\ \hline y = -1 & \frac{1}{8} & \frac{1}{2} \\ y = 0 & 0 & \frac{1}{4} \\ y = 1 & \frac{1}{8} & 0 \\ \end{array} \] Find: (a) The marginal distribution of \(X\). (b) The marginal distribution of \(Y\). (c) The conditional distribution of \(X\) given \(Y = -1\). ### Explanation The table provides a joint probability distribution for random variables \(X\) and \(Y\). The rows correspond to values of \(Y\) (i.e., \(-1\), \(0\), \(1\)) and the columns correspond to values of \(X\) (i.e., \(-1\), \(1\)). Each cell contains the joint probability \(P(X = x, Y = y)\). The tasks are: - **Marginal distribution of \(X\)**: Sum the probabilities over all possible values of \(Y\) for each value of \(X\). - **Marginal distribution of \(Y\)**: Sum the probabilities over all possible values of \(X\) for each value of \(Y\). - **Conditional distribution of \(X\) given \(Y = -1\)**: Use the joint probabilities where \(Y = -1\) and normalize them by the probability of \(Y = -1\).
Transcribed Image Text:### Problem 3.69 Given the values of the joint probability distribution of \(X\) and \(Y\) shown in the table: \[ \begin{array}{c|cc} & x = -1 & x = 1 \\ \hline y = -1 & \frac{1}{8} & \frac{1}{2} \\ y = 0 & 0 & \frac{1}{4} \\ y = 1 & \frac{1}{8} & 0 \\ \end{array} \] Find: (a) The marginal distribution of \(X\). (b) The marginal distribution of \(Y\). (c) The conditional distribution of \(X\) given \(Y = -1\). ### Explanation The table provides a joint probability distribution for random variables \(X\) and \(Y\). The rows correspond to values of \(Y\) (i.e., \(-1\), \(0\), \(1\)) and the columns correspond to values of \(X\) (i.e., \(-1\), \(1\)). Each cell contains the joint probability \(P(X = x, Y = y)\). The tasks are: - **Marginal distribution of \(X\)**: Sum the probabilities over all possible values of \(Y\) for each value of \(X\). - **Marginal distribution of \(Y\)**: Sum the probabilities over all possible values of \(X\) for each value of \(Y\). - **Conditional distribution of \(X\) given \(Y = -1\)**: Use the joint probabilities where \(Y = -1\) and normalize them by the probability of \(Y = -1\).
Expert Solution
Step 1

 

  x=-1 x=1 Total
y=-1 18 12 58
y=0 0 14 14
y=1 18 0 18
Total 14 34 1
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