Let h be a dierentiable nondecreasing function eh x z

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

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

Unformatted text preview: d Y be the number of successes in the last m. By (2.13), X and Y independent. Clearly their sum is binomial(n, p). Formula (A.16) generalizes in the usual way to continuous distributions: regard the probabilities as density functions and replace the sum by an integral. Z fX +Y (z ) = fX (x)fY (z x) dx (A.17) Example A.17. Let U and V be independent and uniform on (0, 1). Compute the density function for U + V . Solution. If U + V = x with 0 x 1, then we must have U x so that V 0. Recalling that we must also have U 0 Zx fU +V (x) = 1 · 1 du = x when 0 x 1 0 If U + V = x with 1 x 2, then we must have U x 1 so that V 1. Recalling that we must also have U 1, Z1 fU +V (x) = 1 · 1 du = 2 x when 1 x 2 x1 Combining the two formulas we see that the density function for the sum is triangular. It starts at 0 at 0, increases linearly with rate 1 until it reaches the value of 1 at x = 1, then it decreases linearly back to 0 at x = 2. A.3 Expected Value, Moments If X has a discrete distribution, then the expected value of h(X ) is X Eh(X...
View Full Document

This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

Ask a homework question - tutors are online