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His and her earnings Researchers randomly select a married couple in which both spouses
Chapter 6, Problem 47(choose chapter or problem)
Researchers randomly select a married couple in which both spouses are employed. Let \(X\) be the income of the husband and \(Y\) be the income of the wife. Suppose that you know the means \(\mu_{\mathrm{X}}\) and \(\mu_{Y}\) and the variances \(\sigma_{\mathrm{X}}^{2}\) and \(\sigma_{\mathrm{Y}}^{2}\) of both variables.
(a) Is it reasonable to take the mean of the total income \(X+Y\) to be \(\mu_{X}+\mu_{Y}\)? Explain your answer.
(b) Is it reasonable to take the variance of the total income to be \(\sigma_{X}^{2}+\sigma_{Y}^{2}\)? Explain your answer.
Questions & Answers
QUESTION:
Researchers randomly select a married couple in which both spouses are employed. Let \(X\) be the income of the husband and \(Y\) be the income of the wife. Suppose that you know the means \(\mu_{\mathrm{X}}\) and \(\mu_{Y}\) and the variances \(\sigma_{\mathrm{X}}^{2}\) and \(\sigma_{\mathrm{Y}}^{2}\) of both variables.
(a) Is it reasonable to take the mean of the total income \(X+Y\) to be \(\mu_{X}+\mu_{Y}\)? Explain your answer.
(b) Is it reasonable to take the variance of the total income to be \(\sigma_{X}^{2}+\sigma_{Y}^{2}\)? Explain your answer.
ANSWER:Step 1 of 3
Define a random variable \(X\) for the income of the husband and \(Y\) for the income of the wife.
Mean and variances for two variable are pairwise stated as, \(\left(\mu_{X}, \sigma_{X}^{2}\right),\left(\mu_{Y}, \sigma_{Y}^{2}\right)\).
The total income is stated using the variable \(X+Y\), say \(Z\).