This content X (in percent) and impurity Y (in percent). LtX 4.2-10. A certain raw mnaterial is classified as to mosture dom variab and Y have the joint pmf given by 3. 4. 2.
This content X (in percent) and impurity Y (in percent). LtX 4.2-10. A certain raw mnaterial is classified as to mosture dom variab and Y have the joint pmf given by 3. 4. 2.
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4.2-10
![**Title: Analyzing Joint Probability Mass Functions**
**Example Problem 4.2-10:**
A certain raw material is classified as to moisture content \(X\) (in percent) and impurity \(Y\) (in percent). Let \(X\) and \(Y\) have the joint probability mass function (pmf) given by the following table:
\[
\begin{array}{c|cccc}
y \backslash x & 1 & 2 & 3 & 4 \\
\hline
2 & 0.10 & 0.20 & 0.30 & 0.05 \\
1 & 0.05 & 0.05 & 0.15 & 0.10 \\
\end{array}
\]
**Tasks:**
(a) Find the marginal pmfs, the means, and the variances.
(b) Find the covariance and the correlation coefficient of \(X\) and \(Y\).
(c) If additional heating is needed with high moisture content and additional filtering with high impurity, such that the additional cost is given by the function \(C = 2X + 10Y^2\) in dollars, find \(E(C)\).
**Explanation of the Table:**
- The table displays the joint pmf for two discrete variables, moisture content \(X\) and impurity \(Y\), measured in percentages.
- The first row and column indicate the possible values for \(y\) and \(x\).
- Each cell within the table gives the probability \(P(X = x, Y = y)\).
**Steps to Solve:**
1. **Marginal PMF**:
- Calculate the marginal pmf for \(X\) by summing the joint probabilities across rows for each \(x\).
- Calculate the marginal pmf for \(Y\) by summing the joint probabilities down columns for each \(y\).
2. **Mean and Variance**:
- Use the marginal pmfs to find the expected values \(\mu_X\) and \(\mu_Y\).
- Calculate the variances \(\sigma_X^2\) and \(\sigma_Y^2\).
3. **Covariance and Correlation**:
- Covariance is calculated using the formula \(\text{Cov}(X, Y) = E(XY) - \mu_X\mu_Y\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F26fe8f9d-a829-4ffe-94fa-74d9058056a0%2Fc35d8aa8-4980-426d-abbd-c2ce82c44e85%2Fjcntiga.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Analyzing Joint Probability Mass Functions**
**Example Problem 4.2-10:**
A certain raw material is classified as to moisture content \(X\) (in percent) and impurity \(Y\) (in percent). Let \(X\) and \(Y\) have the joint probability mass function (pmf) given by the following table:
\[
\begin{array}{c|cccc}
y \backslash x & 1 & 2 & 3 & 4 \\
\hline
2 & 0.10 & 0.20 & 0.30 & 0.05 \\
1 & 0.05 & 0.05 & 0.15 & 0.10 \\
\end{array}
\]
**Tasks:**
(a) Find the marginal pmfs, the means, and the variances.
(b) Find the covariance and the correlation coefficient of \(X\) and \(Y\).
(c) If additional heating is needed with high moisture content and additional filtering with high impurity, such that the additional cost is given by the function \(C = 2X + 10Y^2\) in dollars, find \(E(C)\).
**Explanation of the Table:**
- The table displays the joint pmf for two discrete variables, moisture content \(X\) and impurity \(Y\), measured in percentages.
- The first row and column indicate the possible values for \(y\) and \(x\).
- Each cell within the table gives the probability \(P(X = x, Y = y)\).
**Steps to Solve:**
1. **Marginal PMF**:
- Calculate the marginal pmf for \(X\) by summing the joint probabilities across rows for each \(x\).
- Calculate the marginal pmf for \(Y\) by summing the joint probabilities down columns for each \(y\).
2. **Mean and Variance**:
- Use the marginal pmfs to find the expected values \(\mu_X\) and \(\mu_Y\).
- Calculate the variances \(\sigma_X^2\) and \(\sigma_Y^2\).
3. **Covariance and Correlation**:
- Covariance is calculated using the formula \(\text{Cov}(X, Y) = E(XY) - \mu_X\mu_Y\).
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