ECE302HW2Soln_Sp10 - ECE 302: Homework 2 Prof. Saul Gelfand...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

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

Unformatted text preview: ECE 302: Homework 2 Prof. Saul Gelfand Spring 2010 Due February 5, 2010 Problem 1 a. The error process described is Bernoulli. Let K be the number of errors out of n operations. We have then P [ K = k ] = parenleftbigg n k parenrightbigg p k (1 p ) n- k , for k = 1 ,...,n . b. We have that type 1 errors occur with probability p . Thus, type 1 errors also form a Bernoulli process with parameter p . If we let K 1 be the number of type 1 errors out of n operations, then P [ K 1 = k 1 ] = parenleftbigg n k 1 parenrightbigg ( p ) k (1 p ) n- k , for k 1 = 1 ,...,n . c. Similar to above. Let K 2 be the number of type 2 errors out of n operations. P [ K 2 = k 2 ] = parenleftbigg n k 2 parenrightbigg ((1 ) p ) k (1 (1 ) p ) n- k , for k 2 = 1 ,...,n . d. We first rewrite the joint probability P [ K 1 = k 1 K 2 = k 2 ] = P [ K 1 = k 1 K = k 1 + k 2 ] = P [ K = k 1 | K = k 1 + k 2 ] P [ K = k 1 + k 2 ] . 1 As mentioned in the hint, we will have that if we are given that K = k = k 1 + k 2 errors occur in n operations, then each of the k errors are either type 1 (with probability ) or type 2 (with probability 1 ). Thus, type 1 errors amoung errors are Bernoulli trials. We then have that: P [ K = k 1 | K = k 1 + k 2 ] = parenleftbigg k 1 + k 2 k 1 parenrightbigg k 1 (1 ) k 2 , and P [ K 1 = k 1 K 2 = k 2 ] = P [ K = k 1 | K = k 1 + k 2 ] P [ K = k 1 + k 2 ] = parenleftbigg k 1 + k 2 k 1 parenrightbigg k 1 (1 ) k 2 parenleftbigg n k 1 + k 2 parenrightbigg p k 1 + k 2 (1...
View Full Document

This note was uploaded on 02/07/2011 for the course ECE 302 taught by Professor Gelfand during the Fall '08 term at Purdue University-West Lafayette.

Page1 / 6

ECE302HW2Soln_Sp10 - ECE 302: Homework 2 Prof. Saul Gelfand...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online