A joint PMF of two variables is given by the following table where a=0.1. Y=1 Y=2 X=1 X=2 0.1 0.7 minus a 0.2 a The expected value of X²Y is equal to
A joint PMF of two variables is given by the following table where a=0.1. Y=1 Y=2 X=1 X=2 0.1 0.7 minus a 0.2 a The expected value of X²Y is equal to
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Overview
The table displays the joint probability mass function (PMF) of two discrete random variables, X and Y. The variable "a" in the table is specified as 0.1.
#### PMF Table
| | Y=1 | Y=2 |
|-----|-----------|--------|
| X=1 | 0.1 | a |
| X=2 | 0.7 - a | 0.2 |
### Explanation
- **X and Y Values**:
- For X=1 and Y=1, the probability is 0.1.
- For X=1 and Y=2, the probability is given by "a" (0.1).
- For X=2 and Y=1, the probability is 0.7 minus "a".
- For X=2 and Y=2, the probability is 0.2.
### Calculation of Probabilities
Here, a = 0.1. Substituting "a" in the table:
- For X=1, Y=2: The probability is 0.1.
- For X=2, Y=1: The probability is 0.6 (since 0.7 - 0.1 = 0.6).
### Expected Value Calculation
The expected value of \( X^2Y \) can be calculated using the formula:
\[ E[X^2Y] = \sum (x^2 \cdot y \cdot P(X=x, Y=y)) \]
Substitute the corresponding values from the PMF table to calculate the expected value. Use the following points:
- For (X=1, Y=1): \(1^2 \cdot 1 \cdot 0.1 = 0.1\)
- For (X=1, Y=2): \(1^2 \cdot 2 \cdot 0.1 = 0.2\)
- For (X=2, Y=1): \(2^2 \cdot 1 \cdot 0.6 = 2.4\)
- For (X=2, Y=2): \(2^2 \cdot 2 \cdot 0.2 = 1.6\)
Add all these values to find the expected value of \( X^2Y \):
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Transcribed Image Text:### Overview
The table displays the joint probability mass function (PMF) of two discrete random variables, X and Y. The variable "a" in the table is specified as 0.1.
#### PMF Table
| | Y=1 | Y=2 |
|-----|-----------|--------|
| X=1 | 0.1 | a |
| X=2 | 0.7 - a | 0.2 |
### Explanation
- **X and Y Values**:
- For X=1 and Y=1, the probability is 0.1.
- For X=1 and Y=2, the probability is given by "a" (0.1).
- For X=2 and Y=1, the probability is 0.7 minus "a".
- For X=2 and Y=2, the probability is 0.2.
### Calculation of Probabilities
Here, a = 0.1. Substituting "a" in the table:
- For X=1, Y=2: The probability is 0.1.
- For X=2, Y=1: The probability is 0.6 (since 0.7 - 0.1 = 0.6).
### Expected Value Calculation
The expected value of \( X^2Y \) can be calculated using the formula:
\[ E[X^2Y] = \sum (x^2 \cdot y \cdot P(X=x, Y=y)) \]
Substitute the corresponding values from the PMF table to calculate the expected value. Use the following points:
- For (X=1, Y=1): \(1^2 \cdot 1 \cdot 0.1 = 0.1\)
- For (X=1, Y=2): \(1^2 \cdot 2 \cdot 0.1 = 0.2\)
- For (X=2, Y=1): \(2^2 \cdot 1 \cdot 0.6 = 2.4\)
- For (X=2, Y=2): \(2^2 \cdot 2 \cdot 0.2 = 1.6\)
Add all these values to find the expected value of \( X^2Y \):
\[
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