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55. Rainy days Imagine that we randomly select a day from the past 10 years. Let x be the recorded rainfall on this date at the airport in Orlando Florida, and Y be the recorded rainfall on this date at Disney World just outside Orlando. Suppose that you know the means mu _(X) and mu _(Y) and the variances sigma _(X)^2 and sigma _(Y)^2 of both variables. a. Can we calculate the mean of the to]al rainfall X+Y to be mu _(X)+mu _(Y) ? Explain your answer. b. Can we calculate the variance of the total rainfall to be sigma _(X)^2+sigma _(Y)^2 ? Explain your answer.

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55. Rainy days Imagine that we randomly select a day from the past 10 years. Let x be the recorded rainfall on this date at the airport in Orlando Florida, and Y be the recorded rainfall on this date at Disney World just outside Orlando. Suppose that you know the means mu _(X) and mu _(Y) and the variances sigma _(X)^2 and sigma _(Y)^2 of both variables. a. Can we calculate the mean of the to]al rainfall X+Y to be mu _(X)+mu _(Y) ? Explain your answer. b. Can we calculate the variance of the total rainfall to be sigma _(X)^2+sigma _(Y)^2 ? Explain your answer.

55. Rainy days Imagine that we randomly select a day from the past 10 years. Let x be the recorded rainfall on this date at the airport in Orlando Florida, and Y be the recorded rainfall on this date at Disney World just outside Orlando. Suppose that you know the means mu _(X) and mu _(Y) and the variances sigma _(X)^2 and sigma _(Y)^2 of both variables. a. Can we calculate the mean of the to]al rainfall X+Y to be mu _(X)+mu _(Y) ? Explain your answer. b. Can we calculate the variance of the total rainfall to be sigma _(X)^2+sigma _(Y)^2 ? Explain your answer.

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ElodieVeteran · Tutor for 12 years

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<p> <br />a. Yes, the mean of the total rainfall \(X+Y\) can be calculated to be \(\mu_X + \mu_Y\).<br />b. No, you cannot calculate the variance of the total rainfall to be \(\sigma_X^2 + \sigma_Y^2\) unless \(X\) and \(Y\) are independent.</p>

Explain

<p><br />a. The mean of the total rainfall \(X+Y\):<br />The mean of the sum of two random variables, in this case, the recorded rainfalls \(X\) and \(Y\), can be calculated by summing the means of the individual variables. This is because the expected value operator (which calculates means) is linear. Therefore, if you know the means \(\mu_X\) and \(\mu_Y\) of \(X\) and \(Y\) respectively, the mean of \(X+Y\) is indeed \(\mu_X + \mu_Y\).<br /><br />b. The variance of the total rainfall \(X+Y\):<br />The variance of the sum of two random variables is not simply the sum of their individual variances unless the variables are independent of each other. The variance of the sum is given by:<br />\[\sigma_{X+Y}^{2} = \sigma_X^2 + \sigma_Y^2 + 2 \cdot Cov(X,Y)\]<br />where \(\sigma_X^2\) and \(\sigma_Y^2\) are the variances of \(X\) and \(Y\), and \(Cov(X,Y)\) is the covariance of \(X\) and \(Y\). If \(X\) and \(Y\) are independent, then \(Cov(X,Y) = 0\), and the variance of the sum is indeed the sum of the variances. However, if \(X\) and \(Y\) are not independent (as might be the case with rainfall at two nearby locations), you cannot simply add the variances to get the variance of the total rainfall.</p>
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