{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw3 - Homework Set No 3 Probability Theory(235A Fall 2009...

This preview shows pages 1–2. Sign up to view the full content.

Homework Set No. 3 – Probability Theory (235A), Fall 2009 Posted: 10/13/09 — Due: 10/20/09 1. Let X be an exponential r.v. with parameter λ , i.e., F X ( x ) = (1 - e - λx )1 [0 , ) ( x ). Define random variables Y = b X c := sup { n Z : n x } (“the integer part of X ”) , Z = { X } := X - b X c (“the fractional part of X ”) . (a) Compute the (1-dimensional) distributions of Y and Z (in the case of Y , since it’s a discrete random variable it is most convenient to describe the distribution by giving the individual probabilities P ( Y = n ) , n = 0 , 1 , 2 , . . . ; for Z one should compute either the distribution function or density function). (b) Show that Y and Z are independent. (Hint: Check that P ( Y = n, Z t ) = P ( Y = n ) P ( Z t ) for all n and t .) 2. (a) Let X, Y be independent r.v.’s. Define U = min( X, Y ), V = max( X, Y ). Find expressions for the distribution functions F U and F V in terms of the distribution functions of X and Y . (b) Assume that X Exp( λ ) , Y Exp( μ ) (and are independent as before). Prove that min( X, Y ) has distribution Exp( λ + μ ). Try to give an intuitive explanation in terms of

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

hw3 - Homework Set No 3 Probability Theory(235A Fall 2009...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online