ECE353: Probability and Random Processes. Lecture 7 -Continuous Random Variable

Transcription

1 ECE353: Probability and Random Processes Lecture 7 -Continuous Random Variable Xiao Fu School of Electrical Engineering and Computer Science Oregon State University xiao.fu@oregonstate.edu

2 Continuous Random Variable Let s consider the dart game again... Recall we map the outcomes to scores in the discrete RV case. Suppose now we do the following: ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 1

3 Continuous Random Variable Let s consider the dart game again... Recall we map the outcomes to scores in the discrete RV case. Suppose now we do the following: ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 2

4 Continuous Random Variable Let s consider the dart game again... Recall we map the outcomes to scores in the discrete RV case. Suppose now we do the following: ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 3

5 Continuous Random Variable We are mapping the outcomes to an interval: [0, 2π] Q: How to compute the probability of P [X = x]? Note that x can take any value between 0 and 2π. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 4

6 Let s do the following: Continuous Random Variable Let us call the interval where x lands I(x). We have P [X = x] P [x I(x)] = 1 N, assuming that X is drawn from [0, 2π] in a uniform manner. The above is true for any N; therefore, take N = P [X = x] = 0. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 5

7 Continuous Random Variable Example: we randomly drawn X from [0, 1] uniformly. Then P [X = x 0 ] is P [X = x 0 ] P [x 0 ɛ X x 0 + ɛ] 2ɛ. Note the above takes not =, why? Since the above holds for any ɛ. Taking ɛ 0, we have 0 P [X = x 0 ] lim ɛ 0 2ɛ = 0 ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 6

8 Continuous Random Variable For continuous random variables (coming up shortly), the probability of any specific value (e.g., π) is zero! PMF is a useless tool for continuous random variables! What about CDF? F X (x 0 ) = P [X x 0 ] = 0, x 0 < 0 x 0, x 0 [0, 1] 1, x 0 > 1 CDF is still perfectly useful! ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 7

9 Consider the following: Continuous Random Variable We know P [x 0 < X x 0 + δ] = F X (x 0 + δ) F X (x 0 ). Consider the term P [x 0 < X x 0 + δ], δ which represents probability per unit length, or probability density. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 8

10 Probability Density Function Take limit: lim δ 0 P [x 0 < X x 0 + δ] δ F X (x 0 + δ) F X (x 0 ) = lim δ 0 δ = df X(x) dx x=x0 When the limit exist, we call it the probability density function (PDF) of X. Definition: PDF of continuous random variable X: f X (x) = df X(x) dx The PDF plays the role of PMF. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 9

11 Probability Density Function Properties of PDF: a) f X (x) 0, x. b) F X (x) = x f X(y)dy. c) f X(y)dy = 1 d) P [x 1 X X 2 ] = x 2 x 1 f X (x)dx. Everyone of the above corresponds to a property of the PMF. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 10

12 Probability Density Function Properties of PDF: a) f X (x) 0, x. b) F X (x) = x f X(y)dy. c) f X(y)dy = 1 d) P [x 1 X X 2 ] = x 2 x 1 f X (x)dx. Properties of PMF: a) P X (x) 0, x. b) F X (x) = y x P X(y). c) x S X P X (x) = 1 d) P [x 1 X X 2 ] = x 2 x=x 1 P X (x). ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 11

13 Probability Density Function Example: You are given the PDF of continuous RV X: f X (x) = { Ce αx, x 0 0, o.w. where α > f X (x) x ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 12

14 Probability Density Function Example: You are given the PDF of continuous RV X: where α > 0. f X (x) = { Ce αx, x 0 0, o.w. Q1: find C. f X (x)dx = C 0 = C 1 α e αx 0 e αx dx = C [ 1 α e αx ] 0 = C α (1 0) = C α f X(x)dx = 1 = C = α. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 13

15 Probability Density Function Q2: find F X (1). F X (1) = 1 f X (x)dx = 1 0 αe αx dx = 1 e α Q3: find F X (x). x f X (y)dy = x 0 αe αx dx = 1 e αx, x 0. Hence, F X (x) = { 1 e αx, x 0. 0, o.w. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 14

16 Example: PDF: Probability Density Function f X (x) = { C, x 2 x 1 0, o.w f X (x) x Q1: find C. Q2: find the CDF. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 15

17 Probability Density Function Example: PDF: f X (x) = { C, x 2 x 1 0, o.w. Q1: find C. 1 = f X (x)dx = 1 C 1 ( x 2dx = c 1 ) x 1 = C(0 + 1) = C = 1. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 16

18 Expected Value of Continuous RV Definition: The expected value (or simply expectation) of a continuous random variable X is defined as E[X] = xf X (x)dx. Recall the expectation of discrete RVs: E[X] = x S X xp X (x). ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 17

19 Expected Value of Continuous RV Example: Continuous RV with PDF f X (x) = { 1, x [0, 1] 0, o.w f X (x) x ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 18

20 Expected Value of Continuous RV Example: Continuous RV with PDF f X (x) = { 1, x [0, 1] 0, o.w. E[X] = xf X(x)dx = 1 0 xdx = 1 2 x2 1 0 = 1/2. CDF: F X (x) = x xf X (y)dy =... = 0, x < 0 x, 0 x 1 1, x > 1 ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 19

21 Theorems Theorem: The expected value of Y = g(x) is E[Y ] = g(x)f X (x)dx. Theorem: for any random variable X (both discrete and continuous RVs): a) E[X µ X ] = 0. b) E[aX + b] = ae[x] + b. c) Var[X] = E[X 2 ] µ 2 X. d) Var[aX + b] = a 2 Var[X]. ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 20