Mathematical Expectation of a Random Variable

Transcription

1 Mathematical ectation of a Random Variable

2 Idea: mean value of a random variable Definition: Weighted mean of the values of the random variable P Condition for the eistence: P < Note: Infinite sets can yield aradoes

3 Idea: mean value of a random variable Definition: Weighted mean of the values of the random variable P f ˆ ˆ * P white observation/ comosition of theurn Urn: 3 White 7 Black Observation P comosition of the urn/ whiteobservation * 3

4 Sum of random variables amle: sum of oints of n dice P htt://mathworld.wolfram.com/dice.html 4

5 Mistakes of intuition Intuition corresonds to ratio. Convergence on ratio. Difference gets as bigger! favorable favorable unfavorable favorableunfavorable 5

6 Idea: mean value of a random variable Note that it is robabilistic interretation of the common sense concet of mean. Two samles of the height of a oulation with a real mean of 80 cm. 5 Mean N

7 Idea: mean value of a random variable Note that it is robabilistic interretation of the common sense concet of mean. Note that if all ossible samles are equally robable, they are equivalent. P P N Mean N

8 Idea: center of mass/equilibrium If the robability P is interreted as mass, and the random variable as distance, the mathematical eectation is the center of mass of the object. P 8

9 amle: Roulette What is the ayoff given by the casino? Roulette selects at random a number between and Players can bet on 8,,9,6,4,3,, number. A Bet over k numbers has a robability of success of k/36 and of getting a ayment from the casino or lossing a. ected value is A fair game would imly k k P { Win, Loss} a k a 9

10 amle: Roulette What is the ayoff given by the casino? The roulette has 37 slots to 36 the 0 slot In a Real game the casino the odds are 35: 7: :3 P < { Win, Loss} 0 36 k a k 37 a k 37 a 37 We need to model the variability 0

11 amle: Parking ticket Which is the eected value of not aying the arking ticket? Parking ticket 3 euros Parking fine 50 euros Prob. of getting caught 0.05 P { Win, Loss} 3* *

12 amle: Parking ticket Which is the eected value of not aying the arking ticket? Imortant issues: Subjective value: 3 euros vs. 50 euros P { Win, Loss} Variability of the savings Possible runs of fines 3* *

13 ected value of a geometric random variable Random Varible {Number of trials until a success} P i n 0 i for How do we sum this series? i,,3,l 3 4 L n n 3 3

14 ected value of a geometric random variable How do we sum this series? 4

15 ected value of a geometric random variable How do we sum this series? 5

16 6 ected value of a geometric random variable How do we sum this series? Series Geometric n n k k n n n n L L L

17 7 ected value of a geometric random variable How do we sum this series? Another way Series Geometric b b d d a a d d b b a d d d d k S d d S k k k k

18 ected value of a geometric random variable Random Varible {Number of trials until a success} P i i for i,,3,l n 0 n n L

19 Relation of eectaition and other statistical measures Skew distribution vs. symetric distribution Mode Median Mean Mode Median Mean Full House: The Sread of cellence from Plato to Darwin by Stehen Jay Gould 9

20 0 Proerty of lineality The eectation of a linear combination of random variables is the linear combination of eectations. Y P Y P P Y Y Y Y β α ω ω β ω ω α ω ω β ω α β α β α β α ω ω ω Ω Ω Ω

21 Finding the maimum Suose that n children of differing heights are laced in line at random. You select the first child from the line and walk with her/him along the line until you encounter a child who is taller or until you have reached the end of the line. If you do encounter a taller child, you reeat the rocess. What is the eected value of the number of children selected from the line? Taken fromtijms, Understanding robability

22 Finding the maimum We define the variable as the number of children selected from the line. We will define the indicator variable: i if the i-th child is selected from the line 0 otherwise Now the number of selected children will be: L n

23 Finding the maimum The robability that the ith child is the tallest among the first i children is /i Therefore: i i i 0 for i,, Ln i L ln n L n n n 3

24 ected number of distinct birthdays What is the eected number of distinct birthdays within a randomly formed grou of 00 ersons. We define the random variable i if the birthday is on day i 0 otherwise The number of birthdays is L 3 Taken fromtijms, Understanding robability 4

25 ected number of distinct birthdays What is the eected number of distinct birthdays within a randomly formed grou of 00 ersons. For each day we have: P P i i P The eected number of distinct birthdays is L i 0 5

26 ected number of distinct birthdays For an arbitrary number of ersons. 365 n ected number of distinct birthdays Number of ersons 6

27 Other Proerties If is a non negative random variable, then Also Y then Y 7

28 Other Proerties What do we mean by Y then Y? Y : Ω R :Ω R 0. Y. 0 P PY P 8

29 Caveats of intuition Does the eectation always eist? Mean N Samle mean always eists P P λ λ e! ectation erhas gives an infinite value!!!! P α htt://hysicsweb.org/articles/world/4/7/9/#w40709 The hysics of the Web 9

30 Caveats of intuition Does the eectation always eist? amle: A Cauchy random variable Mean N P P π < < P α htt://hysicsweb.org/articles/world/4/7/9/#w40709 The hysics of the Web 30

31 3 Caveats of intuition Why is infinite? Divergent series Note that for high values of P π α P >> >> π π π

32 3 Caveats of intuition Why is infinite? Value of the harmonic series N > > L L L RPASSAR

33 ected waiting times Geometric Pareto gausian 33

34 ected value of a Binomial 34