{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Prelim_Exam_Solution

# Prelim_Exam_Solution - Exam 1 Solution ORIE 3500/5500...

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

Exam 1 - Solution ORIE 3500/5500: Engineering Probability and Statistics II Question 1 (i) (a) The events A and B are independent if P ( A B ) = P ( A ) P ( B ). If P ( B ) > 0, this is equivalent to P ( A | B ) = P ( A ), i.e. whether or not B has happened, does not influence the likelihood of A . (b) CDF of a random variable X is a function F : R [0 , 1], such that F ( x ) = P ( X x ) . A CDF characterizes the distribution of all random variables, including both discrete and absolutely continuous random variables. (c) PDF of an absolutely continuous random variable Y is a function f : R [0 , ) such that -∞ f ( t ) dt = 1 and P ( Y y ) = y -∞ f ( t ) dt. (d) Expected value of a discrete random variable Z taking values z 1 , z 2 , . . . , z n with prob- abilities p 1 , p 2 , . . . , p n is E ( Z ) = n i =1 z i p i . (ii) (a) Denote by Ω the sample space, F is the family of events. A mapping P : F → R is called a probability law, if the following holds: i. For any event A , P ( A ) 0. ii. P (Ω) = 1. iii. If A 1 , A 2 , . . . are disjoint events, then P ( A 1 A 2 ∪ · · · ) = P ( A 1 ) + P ( A 2 ) + · · · . These axioms can be used in many engineering models to compute the chance of any possible outcome of an experiment. For example if the sample space is finite, and we

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 / 4

Prelim_Exam_Solution - Exam 1 Solution ORIE 3500/5500...

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

View Full Document
Ask a homework question - tutors are online