MATH 20: PROBABILITY Expected Value of Discrete Random - PowerPoint PPT Presentation

MATH 20: PROBABILITY Expected Value of Discrete Random Variables Xingru Chen xingru.chen.gr@dartmouth.edu XC 2020

Important Distributions Hypergeometric Discrete Uniform Distribution Distribution 𝑛 𝜕 = 1 $ &"$ % !"% ℎ 𝑂, 𝑙, 𝑜, 𝑦 = 𝑜 & ! Negative Binomial Binomial Distribution Distribution 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑣 𝑦, 𝑙, 𝑞 𝑙 𝑞 $ 𝑟 !"$ = 𝑦 − 1 𝑙 − 1 𝑞 $ 𝑟 %"$ Geometric Poisson Distribution Distribution 𝑄 𝑌 = 𝑙 = 𝜇 $ 𝑄 𝑈 = 𝑜 = 𝑟 !"# 𝑞 𝑙! 𝑓 "' XC 2020

Average Score of a Pre-employment Psychometric Test poor good 1 2 3 4 5 6 7 8 9 10 10 Scot Sc ott Su Summers Ra Raven Darkh kholme Av Average Ki Kitty Pr Pryde He Henry Ha Hank McCoy Bobby Drake ke Er Eric Leh Lehns nser err Je Jean Gr Grey Em Emma Fr Frost Charles Xa Xavier Ja James Logan Howlett XC 2020

1 2 3 4 5 6 7 8 9 10 10 Average 𝜈 Sc Scot ott Su Summers 5 + 7 + 6 + 2 + 4 + 7 + 9 + 6 + 10 + 2 Ra Raven Darkh kholme 10 = 5.8 Ki Kitty Pr Pryde He Henry Ha Hank McC McCoy 2×2 + 4 + 5 + 2×6 + 2×7 + 9 + 10 = 5.8 Bobby Drake ke 10 Eric Er Leh Lehns nser err Jean Je Gr Grey 5 ×2 + 1 1 10 ×4 + 1 5 ×6 + 1 5 ×7 + 1 10 ×9 Emma Em Fr Frost + 1 Charles Xa Xavier 10 ×10 = 5.8 James Ja Logan Howlett Ho Av Average 𝝂 # frequency×value !∈# XC 2020

Expected Value § Let 𝑌 be a numerica cally-val valued discrete random variable with sample space Ω and distribution function 𝑛(𝑦) . The expected value 𝐹(𝑌) is de fi ned by 𝐹 𝑌 = ∑ !∈$ 𝑦𝑛(𝑦) , provided this sum con olutely . converges abs bsol § The expected value 𝐹(𝑌) is often referred to as the mean, and can be denoted by 𝜈 for short. § If the above sum does not converge absolutely, then we say 𝑌 does not have an expected value. | # 𝑦𝑛 𝑦 | < ∞ ! !∈$ XC 2020

Law of Large Numbers The Law of Large Numbers can help us justify frequency concept of probability and the interpretation of expected value as the average value to be expected in experiments repeated a large number of times. Expect cted value 𝑭(𝒀) # 𝑦𝑛(𝑦) !∈$ Average 𝝂 Av # frequency×value !∈# XC 2020

Example 1 Tos oss a coi coin head or tail 𝑛 𝑦 = 1 2 𝐹 𝑌 = ⋯ XC 2020

Example 2 Rol oll a dice ce 1, 2, 3, 4, 5, or 6 𝑛 𝑦 = 1 6 𝐹 𝑌 = ⋯ XC 2020

Example 3 Suppose that we fl ip a coin until a head fi rst appears, and if the number of tosses equals 𝑜 , then we are paid 𝑜 dollars. What is the expected value of the payment? Expect cted value 𝑭(𝒀) # 𝑦𝑛(𝑦) !∈$ () () () %*' % 𝑜 1 (1 𝑜 1 𝑜𝑟 %*' 𝑞 = # # 2) = # = 2 2 2 %&' %&' %&' (1/2 + 1/4 + 1/8 + ⋯ ) + (1/4 + 1/8 + 1/16 + ⋯ ) + (1/8 + 1/16 + ⋯ ) + ⋯ = 1 + 1/2 + 1/4 + 1/8 + ⋯ = 2 XC 2020

Example 3 (continued) Suppose that we fl ip a coin until a head fi rst appears, and if the number of tosses equals 𝑜 , 2 % dollars. What then we are paid is the expected value of the payment? Expect cted value 𝑭(𝒀) # 𝑦𝑛(𝑦) !∈$ () () () % 2 % 1 2 % 𝑟 %*' 𝑞 = # # = # 1 = +∞ 2 %&' %&' +&' | # 𝑦𝑛 𝑦 | < ∞ ! !∈$ XC 2020

Example 4 Consider the general Bernoulli trial process. As usual, we let 𝑌 = 1 if the outcome is a success and 0 if it is a failure. Expect cted value 𝑭(𝒀) Ber Bernoulli t tria ial # 𝑦𝑛(𝑦) 𝑞, 𝑌 = 1 !∈$ 𝑛 𝑦 = K 1 − 𝑞, 𝑌 = 0 1×𝑞 + 0× 1 − 𝑞 = 𝑞 XC 2020

Binomia Bin ial 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞 , 𝑟 %*, 𝐹 𝑌 = 𝑜𝑞 Ge Geometric 𝐹 𝑌 = 1 𝑄 𝑈 = 𝑜 = 𝑟 %*' 𝑞 𝑞 Po Poisson 𝑄 𝑌 = 𝑙 = 𝜇 , 𝐹 𝑌 = 𝜇 𝑙! 𝑓 *- Negative Ne binomi omial 𝐹 𝑌 = 𝑙 𝑟 𝑣 𝑦, 𝑙, 𝑞 = 𝑦 − 1 𝑙 − 1 𝑞 , 𝑟 !*, 𝑞 Hy Hyper ergeo eomet etric ic , .*, 𝐹 𝑌 = 𝑜 𝑙 ! %*! ℎ 𝑂, 𝑙, 𝑜, 𝑦 = . 𝑂 % XC 2020

Binomial Distribution and Poisson Distribution Bi Binomial Dist stribution 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞 $ 𝑟 !"$ Poisson Po Distribution 𝑄 𝑌 = 𝑙 = 𝜇 $ 𝑙! 𝑓 "' 𝑞 = -/ = % , 𝑢 = 1 , 𝑜 → ∞ XC 2020

Bin Binomia ial 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞 , 𝑟 %*, 𝐹 𝑌 = 𝑜𝑞 Expect cted value 𝑭(𝒀) # 𝑦𝑛(𝑦) !∈$ % % % 𝑙 𝑜 𝑙 𝑜 𝑜! 𝑙 𝑞 , 𝑟 %*, = # 𝑙 𝑞 , 𝑟 %*, = # 𝑙! 𝑜 − 𝑙 ! 𝑞 , 𝑟 %*, # 𝑙 ,&0 ,&' ,&' % % (𝑜 − 1)! 𝑜 − 1 (𝑙 − 1)! 𝑜 − 𝑙 ! 𝑞 ,*' 𝑟 %*, = 𝑜𝑞 # 𝑙 − 1 𝑞 ,*' 𝑟 %*, = # 𝑜𝑞 ,&' ,&' %*' 𝑜 − 1 𝑞 1 𝑟 %*'*1 = 𝑜𝑞 = 𝑜𝑞 # 𝑚 1&0 XC 2020

Po Poisson 𝑄 𝑌 = 𝑙 = 𝜇 , 𝐹 𝑌 = 𝜇 𝑙! 𝑓 *- Expect cted value 𝑭(𝒀) # 𝑦𝑛(𝑦) !∈$ () () 𝑙 𝜇 , 𝑙 𝜇 , 𝑙! 𝑓 *- = # 𝑙! 𝑓 *- # ,&0 ,&' () 𝜇 ,*' (𝑙 − 1)! 𝑓 *- = # 𝜇 ,&' () 𝜇 1 𝑚! 𝑓 *- = 𝜇 = 𝜇 # 1&0 XC 2020

Expectation of Functions of Random Variables § If 𝑌 is a discrete random variable with sample space Ω and distribution function 𝑛(𝑦) , and if 𝜚: Ω → 𝑆 is a function, then 𝐹 𝜚(𝑌) = ∑ !∈$ 𝜚(𝑦)𝑛(𝑦) , provided the series converges absolutely. Expect cted value 𝑭(𝒀) Expect cted value 𝑭 𝝔(𝒀) # 𝑦𝑛(𝑦) # 𝜚(𝑦)𝑛(𝑦) !∈$ !∈$ XC 2020

Example 1 Tos oss a coi coin head or tail 𝑛 𝑦 = 1 2 𝜚 𝑌 = K1, 𝑌 = Head 0, 𝑌 = Tail 𝐹 𝜚 𝑌 = ⋯ XC 2020

Fixed Points § Since a permutation is a one-to-one mapping of the set onto itself, it is of interest to ask how many points (elements) are mapped onto themselves. Such points are called fi fixed po points of the mapping. 2 2 2 1 3 1 2 3 1 3 1 3 2 2 3 1 1 3 XC 2020

Example 2 2 Let 𝑍 be the number of fi xed points in a random permutation of the set {𝑏, 𝑐, 𝑑} . To fi nd the 1 2 3 expected value of 𝑍 , it is helpful to consider the 1 3 basic random variable associated with this experiment, namely the random variable 𝑌 which 1 3 2 represents the random permutation. 2 1 3 Ra Random pe perm rmutation 3! = 6 possible outcomes 2 3 1 𝑛 𝑦 = 1 3 1 2 6 3 2 1 𝐹 𝑍 = ⋯ XC 2020

Example 2 2 Let 𝑍 be the number of fi xed points in a random permutation of the set {𝑏, 𝑐, 𝑑} . To fi nd the 1 2 3 expected value of 𝑍 , it is helpful to consider the 1 3 basic random variable associated with this experiment, namely the random variable 𝑌 which 1 3 2 represents the random permutation. Random Ra pe perm rmutation 2 1 3 3! = 6 possible outcomes 2 3 1 𝑛 𝑦 = 1 6 3 1 2 𝐹 𝑍 = 3 1 6 + 1 1 6 + 1 1 6 + 0 1 6 + 0 1 6 + 1 1 3 2 1 6 = 1 XC 2020

The Sum of Two Random Variables § Let 𝑌 and 𝑍 be random variables with fi nite expected values. Then 𝐹(𝑌 + 𝑍) = 𝐹(𝑌) + 𝐹(𝑍) , and if 𝑑 is any constant, then 𝐹(𝑑𝑌) = 𝑑𝐹(𝑌) . § It can be shown that the expected value of the sum of any fi nite number of random variables is the sum of the expected values of the individual random variables. 𝐹 𝑌 ' + 𝑌 2 + ⋯ + 𝑌 % = 𝐹 𝑌 ' + 𝐹 𝑌 2 + ⋯ + 𝐹(𝑌 % ) . Mu Mutual independence ce of of the summands is not ot ! ne needed. XC 2020

Linearity 𝐹(𝑌 + 𝑍) = 𝐹(𝑌) + 𝐹(𝑍) 𝐹(𝑑𝑌) = 𝑑𝐹(𝑌) . 𝐹(𝑏𝑌 + 𝑐) = 𝑏𝐹(𝑌) + 𝑐 XC 2020

Proof Let the sample spaces of 𝑌 and 𝑍 be denoted by Ω ( and Ω ) , and suppose that Ω ( = 𝑦 # , 𝑦 * , ⋯ Consider the random variable 𝑌 + 𝑍 to be the result of applying the function 𝜚 𝑦, 𝑧 = 𝑦 + 𝑧 to the joint random and variable (𝑌, 𝑍) . Then according to the Expectation of Functions of Random Variables, we have Ω ) = 𝑧 # , 𝑧 * , ⋯ . 𝐹 𝑌 + 𝑍 = ? ? 𝑦 + + 𝑧 $ 𝑄(𝑌 = 𝑦 + , 𝑍 = 𝑧 $ ) . 𝑄(𝑌 = 𝑦 " , 𝑍 = 𝑧 ! ) + $ ! = 𝑄(𝑌 = 𝑦 " ) = ? ? 𝑦 + 𝑄(𝑌 = 𝑦 + , 𝑍 = 𝑧 $ ) + ? ? 𝑧 $ 𝑄(𝑌 = 𝑦 + , 𝑍 = 𝑧 $ ) + $ + $ . 𝑄(𝑌 = 𝑦 " , 𝑍 = 𝑧 ! ) = ? 𝑦 + 𝑄(𝑌 = 𝑦 + ) + ? 𝑧 $ 𝑄(𝑍 = 𝑧 $ ) " + $ = 𝑄(𝑍 = 𝑧 ! ) = 𝐹 𝑌 + 𝐹(𝑍) . XC 2020

Bin Binomia ial 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞 , 𝑟 %*, 𝐹 𝑌 = 𝑜𝑞 Expect cted value 𝑭(𝒀) 3 4 5 1 2 # 𝑦𝑛(𝑦) !∈$ ✗ ✓ ✗ ✗ ✓ 𝑌 + = K1, success 0, failure 2 A single Bernoulli trial 𝐹 𝑌 + = 1×𝑞 + 0× 1 − 𝑞 = 𝑞 ✓ 𝐹 𝑌 𝑜 Bernoulli trials = 𝐹 𝑌 ' + 𝑌 2 + ⋯ + 𝑌 % = 𝑜𝑞 XC 2020