{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

hw05 - EX and Var X Problem 4 Suppose n balls are...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
ORIE 360/560 Fall 2007 Assignment 5 Due Friday, Oct. 5 at 2:00 pm Please remember to show your work! Please remember to put your section number and net id on the assignment! Problem 1. This problem extends the investment example from class. Suppose you are investing $20 , 000 in two mutual funds. Let X be the return of fund 1 and Y the return of fund 2. Suppose EX = EY = μ , but the returns have different variances, Var X = σ 2 X and Var Y = σ 2 Y . Suppose further that the correlation of X and Y is ρ . Find the amount of money a that you should invest in fund 1 (so that $20 , 000 - $ a is invested in fund 2) in order to minimize the resulting variance. Problem 2. A continuous random vector ( X, Y ) has joint pdf f X,Y ( x, y ) = ( 3 x if 0 x 1 and - (1 - x ) y (1 - x ) , 0 otherwise. (a) Find Cov( X, Y ). (b) Are X and Y independent? (c) Find the ρ | X | , | Y | , the correlation of | X | and | Y | . Problem 3. A continuous random variable has density f X ( x ) = λ 1 - e - λ e - λx , for 0 x 1 , where λ > 0. (a) Compute the moment generating function φ X of X . (b) Use your answer to part (a) to compute
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: EX and Var X . Problem 4. Suppose n balls are distributed at random into r boxes. Let X i = 1 if the i th box is empty, X i = 0 otherwise. (a) Compute EX i . (b) Compute EX i X j , for i 6 = j . (c) Let S be the number of empty boxes. Find ES and Var S . Problem 5. Suppose ( X,Y ) is uniformly distributed on the 2-dimensional unit disc D 2 = { ( x,y ) : x 2 + y 2 ≤ 1 } . Find . .. (a) . ..the marginal cdf F X of X . (b) . ..the conditional cdf of Y given X = 1 / 2. The following problem is optional for students enrolled in 360, required for students enrolled in 560. Problem 6. Suppose X is a continuous random variable with pdf f , and suppose that X is bounded. That is, there exists a constant a > 0 such that P [-a ≤ X ≤ a ] = 1. Prove that the moment generating function φ X of X satisfies φ X ( t ) < ∞ for all real t . Hint: first assume t > ....
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern