An urn contains 5 white balls, 3 black balls and 2 red balls. Experiment # 1:...

 

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An urn contains 5 white balls, 3 black balls and 2 red balls. Experiment # 1: A single ball is drawn at random from the urn. Let X be the random variable that is 1 if the ball is black, and is 0 otherwise. Let Y be the random variable that is 1 if the ball is white or black, and is 0 otherwise. (a) Find the variance of X. (b) Find the variance of Y. (c) Find the covariance of X and Y. (Note that the covariance of X and Y is defined as Cov(X, Y) = E[ (X-E(X)) (Y-E(Y))].) (d) Interpret your answer to Part (c). (Do you expect the covariance to be 0 or non-zero? Positive or negative? Why?) Experiment # 2: We keep drawing a single ball at random from the urn, with replacement, until we draw a red ball. (We stop when we draw the red ball.) Let Z be the number of total balls that we draw in Experiment # 2. Let W be the number of red balls that we draw in Experiment # 2. (e) Find the variance of Z. (In addition, find the standard deviation of Z.) (f) Find the variance of W. (In addition, find the standard deviation of W.) (g) Find the covariance of Z and W. (h) Interpret your answer to Part (g). (Is your answer what you would expect intuitively? Why?) An urn contains 5 white balls, 3 black balls and 2 red balls. Experiment # 1: A single ball is drawn at random from the urn. Let X be the random variable that is 1 if the ball is black, and is 0 otherwise. Let Y be the random variable that is 1 if the ball is white or black, and is 0 otherwise. (a) Find the variance of X. (b) Find the variance of Y. (c) Find the covariance of X and Y. (Note that the covariance of X and Y is defined as Cov(X, Y) = E[ (X-E(X)) (Y-E(Y))].) (d) Interpret your answer to Part (c). (Do you expect the covariance to be 0 or non-zero? Positive or negative? Why?) Experiment # 2: We keep drawing a single ball at random from the urn, with replacement, until we draw a red ball. (We stop when we draw the red ball.) Let Z be the number of total balls that we draw in Experiment # 2. Let W be the number of red balls that we draw in Experiment # 2. (e) Find the variance of Z. (In addition, find the standard deviation of Z.) (f) Find the variance of W. (In addition, find the standard deviation of W.) (g) Find the covariance of Z and W. (h) Interpret your answer to Part (g). (Is your answer what you would expect intuitively? Why?)