3-1.1 Two random variables have a joint probability distribution function defined by F(x,y) = 0 = xy = 1 a) Sketch this distribution function. x < 0, y < 0 0≤x≤1, 0 ≤ y ≤1 x > 1, y > 1 b) Find the joint probability density function and sketch it. c) Find the joint probability of the event X ≤ and Y > 3-1.4 Let X be the outcome from rolling one die and Y the outcome from rolling a second die. a) Find the joint probability of the event X ≤ 3 and Y > 3. b) Find E[XY]. c) Find E[] 3-4.3 A random variable X has a variance of 9 and a statistically independent random variable Y has a variance of 25. Their sum is another random variable Z = X + Y. Without assuming that either random variable has zero mean, find a) the correlation coefficient for X and Y b) the correlation coefficient for Y and Z c) the variance of Z. 3-7.1 A random variable X has a probability density function of the form fx(x) = e-xu(x) and an independent random variable Y has a probability density function of fr (y) = 3e-3 u(y) Using characteristic functions, find the probability density function of Z = X +Y.

3-1.1 Two random variables have a joint probability distribution function defined by F(x,y) = 0 = xy = 1 a) Sketch this distribution function. x < 0, y < 0 0≤x≤1, 0 ≤ y ≤1 x > 1, y > 1 b) Find the joint probability density function and sketch it. c) Find the joint probability of the event X ≤ and Y > 3-1.4 Let X be the outcome from rolling one die and Y the outcome from rolling a second die. a) Find the joint probability of the event X ≤ 3 and Y > 3. b) Find E[XY]. c) Find E[] 3-4.3 A random variable X has a variance of 9 and a statistically independent random variable Y has a variance of 25. Their sum is another random variable Z = X + Y. Without assuming that either random variable has zero mean, find a) the correlation coefficient for X and Y b) the correlation coefficient for Y and Z c) the variance of Z. 3-7.1 A random variable X has a probability density function of the form fx(x) = e-xu(x) and an independent random variable Y has a probability density function of fr (y) = 3e-3 u(y) Using characteristic functions, find the probability density function of Z = X +Y.

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter9: Counting And Probability

Section9.4: Expected Value

Problem 20E

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