Expected Value (10D) Young Won Lim 6/12/17
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1 Expected Value (10D)
2 Copyright (c) 2017 Young W. Lim. Permissios granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave.
3 Expected Value Suppose random variable X can take value x 1 with probability p 1, value x 2 with probability p 2, and so on, up to value x k with probability p k. Then the expectation of this random variable X is defined as Since all probabilities pi add up to one ( p 1 + p p k = 1), the expected value can be viewed as the weighted average, with p i s being the weights: Expected Value (10D) 3
4 Arithmetic Mean 2 elements {a, b} 3 elements {a, b, c} A= a b 2 A= a b c 3 n elements {a 1, a 2,..., a n } A= a 1 a 2 a n = 1 n n n i=1 a i Expected Value (10D)
5 Arithmetic Mean - Example = 3 3 =2 3=2 A same length Arithmetic Mean: A = same length 2 6 = =3 =3 A Arithmetic Mean: A = Expected Value (10D) 5
6 Geometric Mean 2 elements {a, b} G= a b a 0, b 0 3 elements {a, b, c} G= 3 a b c a 0, b 0, c 0 n elements {a 1, a 2,..., a n } n G= a n 1 a 2 a n = i=1 1 n a i a i 0 Expected Value (10D) 6
7 Geometric Mean - Example 2 2 = = 2 =G 2 same area Geometric Mean: G = 2 same volume 2 = = 3 =G 3 Geometric Mean: G = Expected Value (10D) 7
8 Harmonic Mean 2 elements a, b 3 elements a, b, c H= 2 1 a 1 b = 2ab a b H= 3 1 a 1 b 1 = 3abc c ab bc ca n elements a 1, a 2,..., a n n H= = n a 1 a 2 a n n 1 i=1 a i Expected Value (10D)
9 Harmonic Mean same distance = D same duration = T A 60 km/hr B 60 km/hr 0 km/hr 0 km/hr A B C Average Speed total distance = total time = 2 D D 60 D = =km/hr 0 Average Speed total distance = total time = 60 T 0 T 2T =50km/hr Harmonic Mean: H = Arithmetic Mean: A = 50 Expected Value (10D) 9
10 Absolute Frequency # of students 0 ~ ~ ~ ~ ~ 90 2 Total Average Score? = 66.5 Expected Value (10D) 10
11 Absolute Frequency & Average - Rectangles x 1 1 student cumulative x 65 x students students 75 x 5 5 students 5 x 2 2 students Expected Value (10D) 11
12 Absolute Frequency & Average Absolute Frequency x 1 1 student cumulative x 20 cumulative x 65 x students students 75 x 5 5 students Avg = x 2 2 students Expected Value (10D) 12
13 Comparison of Histograms Absolute Frequency Group A # of students 0 ~ ~ ~ ~ ~ 90 2 absolute Group A Total Group B # of students 0 ~ ~ ~ ~ 0 9 absolute Group B 0 ~ 90 5 Total Expected Value (10D) 13
14 Normalization Group A (20 students) Total 20 student counts 1.0 Normalization A common scale makes comparison convenient f i = N = k n k f i N relative count (absolute) count total counts = j n j Total 25 student counts Group B (25 students) Expected Value (10D) 1
15 Relative Frequency Normalization f i = N = i Group A f i Group B f i absolute relative 0 ~ /20 = ~ 60 /20 = ~ 70 /20 = ~ 0 5 5/20 = ~ /20 = 0.10 Total 20 20/20 = 1.00 N = i = 20 absolute relative 0 ~ /25 = ~ /25 = ~ 70 /25 = ~ 0 9 9/25 = ~ /25 = 0.20 Total 25 25/25 = 1.00 N = i = i f i = 1.00 i f i = 1.00 Expected Value (10D) 15
16 Comparison of Histograms Relative Frequency Group A f i 0 ~ /20 = ~ 60 /20 = ~ 70 /20 = ~ 0 5 5/20 = ~ /20 = 0.10 Total 20 20/20 = 1.00 Group B absolute absolute N = i = 20 f i relative relative 0 ~ /25 = ~ /25 = ~ 70 /25 = ~ 0 9 9/25 = ~ /25 = 0.20 Total 25 25/25 = 1.00 N = i = 25 relative relative Group A Group B Expected Value (10D) 16
17 Relative Frequency & Average Relative Frequency Group A relative relative cumulative relative cumulative relative x x x.0 1/20 /20 / x.25 5/ Avg = x.10 2/ Expected Value (10D) 17
18 Average Group A absolute f i relative 0 ~ /20 = ~ 60 /20 = ~ 70 /20 = ~ 0 5 5/20 = ~ /20 = 0.10 Total 20 20/20 = 1.00 N = i = 20 absolute Group A relative Group A Average Score? = = = 66.5 Expected Value (10D) 1
19 Accumulative Frequency Group A absolute accumulative absolute 0 ~ ~ ~ ~ ~ Total N = i = 20 accumulative absolute Group A Group A f i relative accumulative relative 0 ~ ~ ~ ~ ~ Total = i f i = 25 accumulative relative Group A Expected Value (10D) 19
20 Relative Frequency & Ratio Absolute Group A f i 2 student absolute relative 0 ~ /20 = ~ 60 /20 = ~ 70 /20 = ~ 0 5 5/20 = ~ /20 = 0.10 Total 20 20/20 = students N = i = 20 students Relative students students Expected Value (10D) 20
21 Normalization Equations (1) f i = N N = f i = N f i relative = count total count total count = count relative freq count = total count relative fraction = part total total = part fraction part = total part 0.xxx = = 0.xxx = 0.xxx Expected Value (10D) 21
22 Normalization Equations (2) f n i N f i = N n N = i f i = N f i 1. 0 f i = slope N i n n f i = slope i f i = slope i 1. 0 N 1. 0 N Expected Value (10D) 22
23 Binomial Probability Distribution n k n k p p p p p q q q q q C (n, k) Expected Value (10D) 23
24 Expected Value of the Binomial Distribution k k P(X=k) ( n k) = n! k!(n k )! ( n 1 k 1) = = n (n 1)! (n k)!k! (n k) = (n 1) (k 1) (n 1)! (k 1)!((n 1) (k 1))! m ( m ) l =0 l pl (1 p) m l = ( p+(1 p)) m Expected Value (10D) 2
25 Expected Value (10D) 25
26 References [1] [2] [3]
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