Example class 10

# Example class 10 - Exercise 1 Let X Y be two discrete...

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The University of Hong Kong Department of Statistics and Actuarial Science STAT1301 Probability and Statistics I Example class 10 Properties of distributions Review Covariance If X and Y are jointly distributed random variables with expectations and , respectively, the covariance of X and Y is Cov(X,Y) = E(XY) – E(X)E(Y) The covariance is the average value of the product of the deviation of X from its mean and the deviation of Y from its mean. Correlation Coefficient If X and Y are jointly distributed random, the correlation of X and Y, denoted by , is The magnitude of Cov (X,Y) is affected by the spread of the distributions of X and Y, the correlation coefficient ρ (X,Y) eliminates this effect by normalizing the covariance. Linear Combination of Random Variables i) Cov(aX, bY) = abCov(X,Y) ii) Cov(X,Y+Z) = Cov(X,Y) + Cov(X,Z) iii) Suppose and , then iv) Var (X + Y) = Var(X) + Var(Y) + 2 Cov(X,Y)

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Unformatted text preview: Exercise 1. Let X; Y be two discrete random variables with their joint mass function f ( x; y ) tabulated below: Y p(X,Y) 1 2 3 4 1 0.11 0.11 0.05 0.10 2 0.11 0.1 0.06 0.10 X 3 0.04 0.07 0.08 0.07 a) Calculate the means of X and Y respectively. b) Calculate the variances of X and Y respectively. c) Calculate the covariance and correlation coefficient between X and Y . 2. In 2006 a hospital sent each surgical patient a bill consisting of X for the services of the hospital and Y for the services of the surgeon. The distributions of X and Y were such that, In 2007 it is expected that the cost of the hospital’s portion of the bill will rise by 10% and the cost of surgical service will rise by 5%. In addition, a service charge of \$200 will be added onto the bill for each surgical patient. Calculate the mean and standard deviation for the bill sent to surgical patients in 2007....
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Example class 10 - Exercise 1 Let X Y be two discrete...

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