MA519_AUG97 - dealt face up from the top of the deck one at...

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

View Full Document Right Arrow Icon
Mathematics 519 Qualifying Exam August 1997 1. A standard deck of playing cards is divided into two stacks, one consisting of the 26 black cards (clubs and spades ), and the other consisting of the 26 red cards (diamonds and hearts ). Each stack is thoroughly shuffled. You are then dealt 5 cards, two from the black stack, and three from the red stack. What is the probability that you are dealt at least one Ace? 2. Let Z, X 1 , X 2 , . . . , X 5 be random variables such that (a) Z has the uniform distribution on the unit interval (0 , 1); and (b) for any z (0 , 1), conditional on Z = z the random variables X 1 , X 2 , . . . , X 5 are independent, identically distributed Bernoulli- z , i.e., for any sequence e 1 , e 2 , . . . , e n of zeros and ones, P ( X i = e i 1 i n | Z = z ) = 5 Y i =1 z e i (1 - z ) 1 - e i . Find P { 5 i =1 X i = 4 } . 3. A standard deck of playing cards is thoroughly shuffled. Cards are then
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: dealt face up from the top of the deck, one at a time, until the first Ace appears. Let Y be the number of cards dealt. Calculate EY. 4. Suppose that the random variables Y,X 1 ,X 2 ,... are independent, and that P { Y = n } = 2-n ∀ n = 1 , 2 ,... P { X k ≥ t } = e-πt ∀ t > 0 and k = 1 , 2 ,.... Let S n = X 1 + X 2 + ··· + X n . Calculate E ( S 3 Y ) . 5. Let Θ 1 , Θ 2 ,... be a sequence of independent, identically distributed ran-dom variables with the uniform distribution on the interval (0 , 2 π ). For n = 1 , 2 ,... define X n = n X k =1 cosΘ k , Y n = n X k =1 sinΘ k , and R 2 n = X 2 n + Y 2 n . Prove that lim n →∞ P { R 2 n ≥ n } exists, and, if possible, evaluate it. 1...
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