ECE353: Probability and Random Processes. Lecture 11- Two Random Variables (II)

Transcription

1 ECE353: Probability and Random Processes Lecture 11- Two Random Variables (II) Xiao Fu School of Electrical Engineering and Computer Science Oregon State University

2 Joint CDF Joint CDF: F X,Y (x, y) = P [X x, Y y]. Joint PDF: Implicitly defined as F X,Y (x, y) = x y u= f X,Y (u, v)dudv and thus f X,Y (x, y) = F X,Y (x, y) x y ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 1

3 Joint CDF Note that for a single RV, computing probabilities of events of interest can be done in two ways: P [2 < X 3] = F X (3) F X (2) and P [2 < X 3] = 3 x=2 The case of two RVs: Now we are in a 2-D space! f X (x)dx. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 2

4 Joint CDF How to compute the probability of B, given F X,Y (x, y)? ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 3

5 Joint CDF How to compute the probability of B, given F X,Y (x, y)? ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 4

6 Joint CDF How to compute the probability of B, given F X,Y (x, y)? ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 5

7 Joint CDF How to compute the probability of B, given F X,Y (x, y)? ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 6

8 Joint CDF How to compute the probability of B, given F X,Y (x, y)? Looks easy? ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 7

9 Joint CDF How to compute the probability of B, given F X,Y (x, y)? ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 8

10 Joint PDF For this case, joint PDF can be very useful: P [(x, y) B] = (x,y) B f X,Y (x, y)dxdy Example: find out P [Y X]. f X,Y (x, y) = { 1, x [0, 1]&y [0, 1] 0, o.w. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 9

11 What is the event B = {Y X}? Joint PDF Solution: 1 x x=0 y=0 1dydx = 1 x=0 xdx = 1 2 x2 1 0 = 1/2 ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 10

12 Some properties of joint PDF: Properties 1. f X,Y (x, y) 0, x, y. 2. f X,Y (x, y)dxdy = 1(= F X,Y (, ) = P [X x, Y y] = 1). 3. f X,Y (x, y)dx = f Y (y). (Marginalization) Proof: F X,Y (, y) = P [x, Y y] = P [Y y] = F Y (y) F X,Y (, y) = = u= y v= y v= f X,Y (u, v)dvdu = q(v)dv q(y) = df Y (y) dy y v= ( = f Y (y) = u= u= ) f X,Y (u, v)du dv f X,Y (u, y)du ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 11

13 Derived RVs Functions of two random variables, i.e., RVs derived from a pair of RVs, W = g(x, Y ), e.g., W = X + Y 2, W = max{x, Y }, W = min{x, Y }, W = X/Y. General problem of interest: Given P X,Y (x, y), F X,Y (x, y) or f X,Y (x, y) and a mapping g(x, y) w. Find P W (w) or F W (w) or f W (w), whichever is appropriate. Discrete RVs: EASY! P W (w) = (x,y) g(x,y)=w P X,Y (x, y) ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 12

14 Derived RVs Continuous RVs: typically first find the CDF of W F W (w) = P [W w] = (x,y) g(x,y) w f X,Y (x, y)dxdy. in general more complicated; but there are some special cases that are easy. Example: W = max{x, Y }. Given F X,Y (x, y), find F W (w). F W (w) = P [W w] = P [max{x, Y } w] = P [X w, Y w] = F X,Y (w, w). ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 13

15 Derived RVs Example: V = min{x, Y }. Given F X,Y (x, y), find F W (w). F V (v) = P [V v] = P [min{x, Y } v] = P [X v or Y v]. which is hard to compute. However, if we consider 1 F V (v) = P [V v] = P [min{x, Y } v] = P [X v, Y v]. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 14

16 Derived RVs Example: V = min{x, Y }. 1 F V (v) = P [V v] = P [min{x, Y } v] = P [X v, Y v]. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 15

17 Derived RVs Example: Roll a pair of independent fair dice, X, Y. Find the PMF of W = X + Y and V = max{x, Y }. Every grid has a probability of 1/36, since we have two independent fair dice (P [X = x, Y = y] = P [X = x]p [Y = y]). ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 16

18 Derived RVs P W (w) = (x,y) x+y=w 1 36, e.g., P W (w) = P [W = 5] = 4/36. Q: what is P W (2)? ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 17

19 Derived RVs What is the PMF of V = max{x, Y }? E.g., P [V = 3] = P [X = 3 or Y 3 X 3 or Y = 3] = 5/36. P [V = 1] = 1/36, P [V = 3] = 5/36, P [V = 6] = 11/36,... increasing! ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 18