Mterm2pa_09k - Practice Midterm Exam Questions - Second...

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Practice Midterm Exam Questions - Second Midterm Chapter 5 Questions. 1. Using the distribution in Question 5-12, part c) (p168) a) Work out the mean and variance of (X+Y). The calculation is in the table. The mean is 2.7. b) Work out the mean and variance of X, then the variance of Y. In the second and third panels of the spreadsheet. The variances are 0.41 and 0.24 respectively. c) What is the relationship between the answers in a) and b)? (Note: This is simply done using the spreadsheet, where you can set up tables for each distribution and use the copy command and the fill down command to quickly replicate similar instructions). (X+Y) P(X+Y) (X+Y)*P(X+Y) (X+Y)-2.7 (X+Y-2.7)^2 g(x,y)*p(x,y) 1 0.06 0.06 -1.7 2.89 0.1734 2 0.34 0.68 -0.7 0.49 0.1666 3 0.44 1.32 0.3 0.09 0.0396 4 0.16 0.64 1.3 1.69 0.2704 2.7 0.65 (Mean) X P(X) X*P(X) X-1.3 (X-1.3)^2 g(x,y)*p(x,y) 0 0.1 0 -1.3 1.69 0.169 1 0.5 0.5 -0.3 0.09 0.045 2 0.4 0.8 0.7 0.49 0.196 1.3 0.41 (Mean) Y P(Y) Y*P(Y) Y-1.4 (Y-1,4)^2 g(x,y)*p(x,y) 1 0.6 0.6 -0.4 0.16 0.096 2 0.4 0.8 0.6 0.36 0.144 1.4 0.24 (mean) We notice that 0.41 + 0.24 = 0.65, verifying our rule for independent random variables that var(X) + var(Y) = var(X+Y). 2. Suppose that the random variable X measures number of hamburgers purchased in a restaurant in an hour and Y measures the number of packages of antacid purchased in the convenience market next door in the same hour. The correlation between the two variables is 0.75. (a) Interpret what this correlation tells us about the relationship between the number of hamburgers purchased and the number of antacids purchased (be thorough in your answer). The correlation measures on the interval -1 to 1 the strength of the (linear) relationship between X and Y. A value of zero means that we will often see large X with small Y and large X with large Y (and vice versa) and a value closer to 1 means we are more likely to see large X with large Y. We have a correlation of 0.75 which suggests that the two random variables are fairly strongly positively correlated, that is we expect to see stronger antacid sales when hamburger sales are high.
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(b) Suppose the variance of X and the variance of Y are both equal to 16. What is the covariance between X and Y. Using the formula for the correlation ρ σ XY XY X Y = we have that XY XY X Y = = = 0 75 4 4 12 . * * (c) What is the variance of (X + Y)? Again, the formula for the variance of (X+Y) we derived in class is var(X+Y)=var(X)+var(Y)+2cov(X,Y) = 16 + 16 +2*12 = 56. 3. The joint distribution of two random variables X and Y is given by Y X 1 2 1 0.05 0.05 2 0.5 0.3 3 0.05 0.05 (a) What is the marginal distribution of X? Y p(X) X 1 2 1 0.05 0.05 0.1 2 0.5 0.3 0.8 3 0.05 0.05 0.1 (b) What is the mean of X? E X
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Mterm2pa_09k - Practice Midterm Exam Questions - Second...

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