Question

7. Your roommate plays video games for anywhere up to two hours on Wednesday nights and for up to one hour on Thursday nights. An expert on "the video game playing habits of roommates" claims that if we let X represent the random variable for how many hours a roommate will play on Wednesday nights of any given week, and Y represents the how many the roommate will play on Thursday night of the same week, then the pdf will be based on the following formula x-x,y, 0 x 2,0 y 1 otherwise Based on the assumption that this expert is right, answer the following a.) Demonstrate that this function satisfies the two properties of a valid joint probability density function b.) What is fx(x), the marginal pdf for X? b.) What is f(y), the marginal pdf for Y? c) What is the probability that your roommate will play video games for no more than one hour on Wednesday and no more than a half hour on Thursday of next week? d.) What is the probability that your roommate will play video games for a total of no more than two hours on Wednesday and Thursday of next week? e.) State the conditional pdf frx(yx) f.) Suppose your roommate plays video games for exactly one hour Wednesday night, what is the probability he or she will play for more than a half hour on Thursday evening of the same week? g.) What are the two means of the marginal distributions and μ? h.) What are the two standard deviations of the marginal distributions ox and ơy? i.) What is E(XY)? j) What is the covariance Cov(X, Y)? k.) What is the correlation coefficient pxy? What does this tell us about the relationship between the two random variables? I.) Are X and Y independent random variables? How do you know? What does this tell us?

Please explain how did you came up with the answer, show work and formulas. I will give you five full stars if you do so.Thank you so much!

Answer 1

子(7,3) : otheraise 手(Χ.3) (i) > @ し( ズーχ 4 (4-43) dy arginal o 2.

3 北可ー廾. 203 2.4+