MATH 20: PROBABILITY Generating Functions Xingru Chen - PowerPoint PPT Presentation

MATH 20: PROBABILITY Generating Functions Xingru Chen xingru.chen.gr@dartmouth.edu XC 2020

di distri ribution Random Variable Password Lo Log I In Forget Password XC 2020

di distri ribution Random Variable Expected Value & Variance Lo Log I In Forget Password XC 2020

Binomial Distribution and Normal Distribution Binomia Bin ial D Dist istrib ibutio ion Norma Nor mal Distribution on 𝐹(𝑌) & 𝑊(𝑌) 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 1 𝑙 𝑞 ! 𝑟 "#! 𝑜𝑞 = 𝜈 𝑓 # %#& ! /() ! § 𝑔 $ 𝑦 = 𝑜𝑞 1 − 𝑞 = 𝜏 ( 2𝜌𝜏 § XC 2020

di distri ribution Random Variable Moments Lo Log I In Forget Password XC 2020

GENERATING FUNCTIONS discrete distribution XC 2020

Moments § If 𝑌 is a random variable with range 𝑦 * , 𝑦 ( , ⋯ of at most countable size, and the distribution function 𝑞 = 𝑞 $ , we introduce the moments of 𝑌 , which are numbers defined as follows: 𝜈 ! = 𝑙 th moment of 𝑌 = 𝐹 𝑌 ! ! 𝑞(𝑦 + ) , -. 𝑦 + = ∑ +,* provided the sum converges. Here 𝑞 𝑦 + = 𝑄(𝑌 = 𝑦 + ) . ? 𝜈 ! = ⋯ XC 2020

𝑙 th moment of 𝑌 𝜈 $ = 1 = &' " 𝑞(𝑦 # ) 𝜈 " = 𝐹 𝑌 " = * 𝑦 # #$% &' 𝜈 ! = 𝐹 1 = * 𝑞(𝑦 # ) 𝜈 % = ⋯ ? #$% &' 𝜈 % = 𝐹 𝑌 = * 𝑦 # 𝑞(𝑦 # ) #$% &' 𝜈 & = ⋯ 𝜈 ( = 𝐹 𝑌 ( = * ( 𝑞(𝑦 # ) ? 𝑦 # #$% XC 2020

Expected Value & Variance 𝑙 th moment of 𝑌 𝜈 " = 𝐹 𝑌 " &' " 𝑞(𝑦 # ) = * 𝑦 # Expe pecte ted Value ue #$% 𝜏 & = ⋯ 𝜈 % = 𝐹 𝑌 &' = * 𝑦 # 𝑞(𝑦 # ) 𝜈 = ⋯ Variance Va #$% 𝜈 ( = 𝐹 𝑌 ( &' ( 𝑞(𝑦 # ) = * 𝑦 # #$% XC 2020

Expected Value & Variance 𝑙 th moment of 𝑌 𝜈 " = 𝐹 𝑌 " &' " 𝑞(𝑦 # ) = * 𝑦 # Expe pecte ted Value ue #$% 𝜏 & = 𝜈 & − 𝜈 % & 𝜈 % = 𝐹 𝑌 &' = * 𝑦 # 𝑞(𝑦 # ) 𝜈 = 𝜈 % Va Variance #$% 𝜈 = 𝜈 % 𝜈 ( = 𝐹 𝑌 ( &' ( 𝑞(𝑦 # ) = * 𝑦 # #$% XC 2020

Moment Generating Functions § We introduce a new variable 𝑢, and de fi ne a function 𝑕(𝑢) as follows: Ex Expect cted val alue 𝑭 𝝔(𝒀) &' 𝑓 !( ! 𝑞(𝑦 # ) . 𝑕 𝑢 = 𝐹 𝑓 !" = ∑ #$% & 𝜚(𝑦)𝑛(𝑦) § We call 𝑕(𝑢) the moment generating function for 𝑌 , and think of it as a convenient bookkeeping device for "∈$ describing the moments of 𝑌 . &' 𝑌 " 𝑢 " &' 𝐹(𝑌 " )𝑢 " &' 𝜈 " 𝑢 " Taylor 𝑕 𝑢 = 𝐹 𝑓 )* = 𝐹 = * = * = * Expansion 𝑙! 𝑙! 𝑙! "$! "$! "$! XC 2020

Moment Generating Functions § If we differentiate 𝑕(𝑢) 𝑜 times and then set 𝑢 = 0 , we get 𝜈 " . &' 𝜈 " 𝑢 " &' 𝑕 𝑢 = 𝐹 𝑓 )* = * 𝑓 )+ ! 𝑞(𝑦 # ) = * = 𝑙! #$% "$! &' 𝜈 " 𝑢 " &' 𝑙! 𝜈 " 𝑢 "-, &' 𝜈 " 𝑢 "-, &' 𝜈 " 𝑢 "-, 𝑒𝑢 , 𝑕 𝑢 = 𝑒 , 𝑒 , 𝑒 , 𝑒𝑢 , * = * 𝑙! 𝑙 − 𝑜 ! = * 𝑒𝑢 , 𝑕 𝑢 | )$! = * 𝑙 − 𝑜 ! | )$! = 𝜈 , 𝑙! 𝑙 − 𝑜 ! "$! "$, "$, "$, 𝑕 , 0 = 𝑒 , = 𝑒𝑢 , 𝑕 𝑢 | )$! = 𝜈 , XC 2020

Uniform Distribution Ra Random vari riable le range: 1, 2, 3, ⋯ , 𝑜 𝑞 * 𝑘 = % distribution function: , Generating funct ction on , 1 *+ 𝜈 , 𝑢 , *+ 𝑜 𝑓 )# = 1 𝑜 𝑓 ) + 𝑓 () + 𝑓 .) + ⋯ + 𝑓 ,) 𝑕 𝑢 = 𝐹 𝑓 %& = & 𝑕 𝑢 = * 𝑓 %" ! 𝑞(𝑦 ' ) = & = 𝑙! '() ,(- #$% / " (/ #" -%) ,(/ " -%) . = 𝑕 , 0 = 𝑒 , = 𝑒𝑢 , 𝑕 𝑢 | )$! = 𝜈 , XC 2020

Uniform Distribution Random Ra vari riable le range: 1, 2, 3, ⋯ , 𝑜 % distribution function: 𝑞 * 𝑘 = , Generating funct ction on / " (/ #" -%) , 𝑓 )# = % , ,(/ " -%) . 𝑕 𝑢 = ∑ #$% *+ 𝜈 , 𝑢 , *+ 𝑕 𝑢 = 𝐹 𝑓 %& = & 𝑓 %" ! 𝑞(𝑦 ' ) = & = 𝑙! Mom Moments '() ,(- 𝜈 % = 𝑕 2 0 = % ,&% ( . , 1 + 2 + 3 + ⋯ + 𝑜 = 𝑕 , 0 = 𝑒 , 𝜈 ( = 𝑕 22 0 = , 1 + 4 + 9 + ⋯ + 𝑜 ( = % (,&%)((,&%) . = 𝑒𝑢 , 𝑕 𝑢 | )$! = 𝜈 , 3 XC 2020

Uniform Distribution Ra Random vari riable le range: 1, 2, 3, ⋯ , 𝑜 % distribution function: 𝑞 * 𝑘 = , Generating funct ction on / " (/ #" -%) , 𝑓 )# = % , ,(/ " -%) . 𝑕 𝑢 = ∑ #$% Mom Moments 𝜈 % = 𝑕 2 0 = % ,&% ( . , 1 + 2 + 3 + ⋯ + 𝑜 = 𝜈 ( = 𝑕 22 0 = , 1 + 4 + 9 + ⋯ + 𝑜 ( = % (,&%)((,&%) . 3 𝜈 = 𝜈 * Expect cted value & variance ce 𝜏 ( = 𝜈 ( − 𝜈 * ( 𝜈 = 𝜈 % = 𝑜 + 1 . 2 , $ -% 𝜏 ( = 𝜈 ( − 𝜈 % ( = %( . XC 2020

Binomial Distribution Random Ra vari riable le range: 0, 1, 2, 3, ⋯ , 𝑜 , distribution function: 𝑞 * 𝑘 = # 𝑞 # 𝑟 ,-# Generating funct ction on , *+ 𝜈 , 𝑢 , *+ 𝑓 )# 𝑜 𝑘 𝑞 # 𝑟 ,-# = 𝑕 𝑢 = 𝐹 𝑓 %& = & 𝑕 𝑢 = * 𝑓 %" ! 𝑞(𝑦 ' ) = & = 𝑙! '() ,(- #$% # (𝑞𝑓 ) ) # 𝑟 ,-# = (𝑞𝑓 ) + 𝑟) , . , , ∑ #$% 𝑕 , 0 = 𝑒 , = 𝑒𝑢 , 𝑕 𝑢 | )$! = 𝜈 , XC 2020

Binomial Distribution Ra Random vari riable le range: 0, 1, 2, 3, ⋯ , 𝑜 , distribution function: 𝑞 * 𝑘 = # 𝑞 # 𝑟 ,-# Generating funct ction on # 𝑞 # 𝑟 ,-# = (𝑞𝑓 ) + 𝑟) , . , , 𝑓 )# 𝑕 𝑢 = ∑ #$% *+ 𝜈 , 𝑢 , *+ 𝑕 𝑢 = 𝐹 𝑓 %& = & 𝑓 %" ! 𝑞(𝑦 ' ) = & = 𝑙! Moments Mom '() ,(- 𝜈 % = 𝑕 2 0 = 𝑜(𝑞𝑓 ) + 𝑟) ,-% 𝑞𝑓 ) | )$! = 𝑜𝑞 . 𝜈 ( = 𝑕 22 0 = 𝑜 𝑜 − 1 𝑞 ( + 𝑜𝑞 . 𝑕 , 0 = 𝑒 , = 𝑒𝑢 , 𝑕 𝑢 | )$! = 𝜈 , XC 2020

Binomial Distribution Ra Random vari riable le range: 0, 1, 2, 3, ⋯ , 𝑜 , distribution function: 𝑞 * 𝑘 = # 𝑞 # 𝑟 ,-# Generating funct ction on # 𝑞 # 𝑟 ,-# = (𝑞𝑓 ) + 𝑟) , . , , 𝑓 )# 𝑕 𝑢 = ∑ #$% Mom Moments 𝜈 % = 𝑕 2 0 = 𝑜𝑞 . 𝜈 ( = 𝑕 22 0 = 𝑜 𝑜 − 1 𝑞 ( + 𝑜𝑞 . 𝜈 = 𝜈 * Expect cted value & variance ce 𝜏 ( = 𝜈 ( − 𝜈 * 𝜈 = 𝜈 % = 𝑜𝑞. ( 𝜏 ( = 𝜈 ( − 𝜈 % ( = 𝑜𝑞(1 − 𝑞) . XC 2020

Geometric Distribution Ra Random vari riable le range: 1, 2, 3, ⋯ , 𝑜 distribution function: 𝑞 * 𝑘 = 𝑟 #-% 𝑞 *+ 𝜈 , 𝑢 , *+ 𝑕 𝑢 = 𝐹 𝑓 %& = & = 𝑓 %" ! 𝑞(𝑦 ' ) = & 𝑙! Generating funct ction on '() ,(- 4/ " , %-5/ " . 𝑓 )# 𝑟 #-% 𝑞 = 𝑕 𝑢 = ∑ #$% Moments Mom 𝑕 , 0 = 𝑒 , = 𝑒𝑢 , 𝑕 𝑢 | )$! = 𝜈 , 4/ " 𝜈 % = 𝑕 2 0 = % 4 . (%-5/ " ) $ | )$! = 4/ " &45/ $" 𝜈 ( = 𝑕 22 0 = %&5 4 $ . (%-5/ " ) % | )$! = Expect cted value & variance ce 𝜈 = 𝜈 * 𝜈 = 𝜈 % = 1 𝜏 ( = 𝜈 ( − 𝜈 * ( 𝑞 . 𝜏 ( = 𝜈 ( − 𝜈 % ( = 5 4 $ . XC 2020

Poisson Distribution Ra Random vari riable le range: 0, 1, 2, 3, ⋯ , 𝑜 𝑞 * 𝑘 = 𝑓 -6 6 ! distribution function: #! Generating funct ction on 𝑓 )# 𝑓 -6 6 ! (6/ " ) ! #! = 𝑓 -6 ∑ #$% = 𝑓 6(/ " -%) . , , 𝑕 𝑢 = ∑ #$% #! Mom Moments 𝜈 % = 𝑕 2 0 = 𝑓 6(/ " -%) 𝜇𝑓 ) | )$! = 𝜇 . 𝜈 ( = 𝑕 22 0 = 𝑓 6(/ " -%) 𝜇 ( 𝑓 () + 𝜇𝑓 ) | )$! = 𝜇 ( + 𝜇 . *+ 𝜈 , 𝑢 , *+ 𝑕 𝑢 = 𝐹 𝑓 %& = & 𝑓 %" ! 𝑞(𝑦 ' ) = & = 𝑙! '() ,(- Expect cted value & variance ce 𝜈 = 𝜈 % = 𝜇. 𝜏 ( = 𝜈 ( − 𝜈 % ( = 𝜇. 𝑕 , 0 = 𝑒 , = 𝑒𝑢 , 𝑕 𝑢 | )$! = 𝜈 , XC 2020

Unifor Un orm 𝑄 𝑌 = 𝑙 = 1 𝑊 𝑌 = " ! #* 𝐹 𝑌 = "-* ( , 𝑜 *( Bin Binomia ial 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝐹 𝑌 = 𝑜𝑞 , 𝑊 𝑌 = 𝑜𝑞𝑟 𝑙 𝑞 ! 𝑟 "#! Geometric Ge 𝐹 𝑌 = * 𝑊 𝑌 = *#/ / , 𝑄 𝑈 = 𝑜 = 𝑟 "#* 𝑞 / ! Po Poisson 𝑄 𝑌 = 𝑙 = 𝜇 ! 𝐹 𝑌 = 𝜇 , 𝑊 𝑌 = 𝜇 𝑙! 𝑓 #0 XC 2020

Unifor Un orm 𝑕 𝑢 = 𝑓 1 (𝑓 "1 − 1) 𝑄 𝑌 = 𝑙 = 1 𝑜(𝑓 1 − 1) 𝑜 Bin Binomia ial 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑕 𝑢 = (𝑞𝑓 1 + 𝑟) " 𝑙 𝑞 ! 𝑟 "#! Geometric Ge 𝑞𝑓 1 𝑄 𝑈 = 𝑜 = 𝑟 "#* 𝑞 𝑕 𝑢 = 1 − 𝑟𝑓 1 Poisson Po 𝑄 𝑌 = 𝑙 = 𝜇 ! 𝑕 𝑢 = 𝑓 0(3 . #*) 𝑙! 𝑓 #0 XC 2020

XC 2020

Po Poisson 𝑄 𝑌 = 𝑙 = 𝜇 ! 𝑕 𝑢 = 𝑓 0(3 . #*) . 𝑙! 𝑓 #0 𝑕 , 0 = 𝑒 , = 𝑒𝑢 , 𝑕 𝑢 | )$! = 𝜈 , 𝐹 𝑌 . = 𝜈 . = 𝑒 . 𝑒𝑢 . 𝑕 𝑢 | )$! = 𝑒 . 𝑒𝑢 . 𝑓 6(/ " -%) | )$! = 𝑒 𝑒𝑢 𝑓 6(/ " -%) 𝜇 ( 𝑓 () + 𝜇𝑓 ) | )$! = 𝑓 6(/ " -%) 𝜇 . 𝑓 .) + 3𝜇 ( 𝑓 () + 𝜇𝑓 ) | )$! = 𝜇 . + 3𝜇 ( + 𝜇 XC 2020