1036: Probability & Statistics

Transcription

1 1036: Probabilit & Statistics Lecture 4 Mathematical pectation Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-1

2 Mean o a Random Variable Let be a random variable with probabilit distribution. The mean or epected value o is ample: d i is discrete and i is continuous I coins are tossed 16 times. The outcomes are 0 head: 4 times; 1 head: 7 times; heads: 5 times. The average number o heads per toss? Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw

3 ample 4.3 Let be the random variables that denotes the lie in hours o a certain electronic device. The probabilit densit unction is 0000 > eleswhere Find the epected lie o this tpe o device. Solution 0000 d Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-3

4 pectation o g Let be a random variable with probabilit distribution. The mean or epected value o the random variable g is [ g ] [ g ] g I is discrete g g : is a RV with pd: g The mean o g43??? d 3 0 I is continuous -1< < elsewhere [ 4 3] g d 4 3 d 8 3 Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 1 4-4

5 pectation o g Let and be two random variables with joint probabilit distribution. The mean or epected value o the random variable g is [ g ] [ g ] g I are discrete g g : Find / or the densit unction g dd 0 < < 0 < elsewhere < 1 I are continuous Sol: dd 5 8 Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-5

6 ample Let and be random variables with joint densit unction 4 0 < < 10 < < 1 0 elsewhere. Find the epected value o Z Solution: Z z dd 1 4dd Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-6

7 Remar g discrete dd g d continuous dd h d h discrete continuous & calculated b joint pd or marginal pd Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-7

8 Wh Variance To measure the variabilit!! The mean o a random variable in statistics describes where the probabilit distribution is centered. The mean does not give adequate description o the shape o the distribution. Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-8

9 Variance and Standard Deviation Let be a random variables with probabilit distribution and mean. The variance o is [ ] [ ] d I is discrete I is continuous The positive square root o the variance is called the standard deviation o The quantit is called the deviation o an observation rom its mean. Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-9

10 ample 4.8 Let a random variable represent the number o automobiles that are used or oicial business purpose on an given weeda. We have the ollowing distributions: or compan A Compan A [ ] Compan B: 1 [ ] 1. 6 or compan B Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-10

11 Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-11 Variance [ ] Proo: This oten simpliies the calculation

12 ample 4.10 is a random variable with the pd 0 1 Mean & variance? 1 < < elsewhere. d d d 5 3 [ ] 1/ 18 Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-1

13 Remars Let be a random variable with probabilit distribution. The variance o the random variable g is g [ ] [ g ] g g g [ ] g g g [ g g ] d I is discrete I is continuous Since g is itsel a random variable with mean g Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-13

14 ample 4.11 Calculate the variance o g3 where is a random variable with probabilit distribution: Solution 0 1/4 1 1/8 1/ 3 1/8 3 6 g [ ] 3 g g g 0 Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-14

15 Covariance Let and be random variables with joint probabilit distribution. The covariance o and is [ ] I discrete [ ] dd I continuous When and are statisticall independentl 0. The inverse is not generall true The sign o the covariance indicates whether the relationship between two dependent random variables is positive or negative. Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-15

16 Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-16 Covariance Proo: This oten simpliies the calculation

17 ample 4.14 The raction o male runners and the raction o emale runners who compete in marathon races is described b the joint densit unction 8 0 Find the covariance o and 0 10 elsewhere Sol d g d 0 elsewhere g h d 4 1 h d elsewhere 4 / 55 Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-17

18 Correlation Coeicient The covariance does not indicate anthing regarding the strength o the relationship since the value is not scale ree which depends on the units measured or both and. Let and be random variables with covariance and standard deviations and respectivel. The correlation coeicient and is 1 ρ 1 ρ ρ 0 ±1 ρ & independent & linear dependence i.e.ab Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-18

19 Linear Combinations o RVs. I a and b are constant then a ba b Proo. I b is constant then bb I a is constant then aa The epected value o the sum or dierence o two or more unctions o a random variable is the sum or dierence o the epected values o the unctions. [ g ± h ] [ g ] [ h ] ± Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-19

20 Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-0 Linear Functions o RVs The epected value o the sum or dierence o two or more unctions o random variable and is the sum or dierence o the epected values o the unctions. [ ] [ ] [ ] h g h g ± ± [ ] [ ] [ ] [ ] h g dd h dd g dd h g h g ± ± ± ± ] [ ] [ ] [ h g h g ± ±

21 Theorem 4.8 Let and be two independent random variables. Then dd h g dd Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-1

22 Linear Combinations o RVs. I a and b are constant then Proo. a b a I b is constant then I a is constant then b a a I and are random variables with joint probabilit distribution then a b ± ab a ± b Proo a ± b {[ a ± b a ± b ] } {[ a a a ± b ] } b b Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw ± ab[ ± ab ] 4-

23 Remars I and are independent then 0 I and are independent then a ± b a b I 1 n are independent a a L a a1 a L 1 1 n n 1 n a n Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-3

24 Variance? I a random variable has a small variance we would epect most o the values to be grouped around the mean. A large value o indicates a greater variabilit and thereore we would epect the spread distribution. Since the total area under a probabilit distribution curve is 1 the area between an two numbers is then the probabilit o the random variable assuming a value between these numbers. Chebshev discovered that the raction o the area between an two values smmetric about the mean is related to the standard deviation. Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-4

25 Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw That is P d d d d Chebshev s Theorem The probabilit that an random variable will assume a value within standard deviations o the mean is at least 11/. That is Proo 1 1 P < irst consider the probabilit We P d d d d d

26 Chebshev s Theorem ¾ or more o the observations o an distribution lie in the interval ± 1-1/ ¾ lower bound onl wea result The use o Chebshev s theorem is relegated to situations where the orm o the distribution is unnown. ample 4. a RV with mean o 8 and variance o 9 1 a. P 4 < < 0 P[ 8 43 < < 8 43] 1 4 b. P P 8 < 6 1 P < 8 < 6 1 P 8 3 < < / 4 Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-6