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

MATH 20: PROBABILITY

Generating Functions Xingru Chen xingru.chen.gr@dartmouth.edu

XC 2020

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Random Variable Password

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Random Variable Expected Value & Variance

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Binomial Distribution and Normal Distribution

Bin Binomia ial D Dist istrib ibutio ion 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞!𝑟"#! Nor Norma mal Distribution

• n

𝑔

$ 𝑦 =

1 2𝜌𝜏 𝑓# %#& !/()! 𝐹(𝑌) & 𝑊(𝑌) § 𝑜𝑞 = 𝜈 § 𝑜𝑞 1 − 𝑞 = 𝜏(

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Random Variable Moments

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GENERATING FUNCTIONS

discrete distribution

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Moments

§ If 𝑌 is a random variable with range 𝑦*, 𝑦(, ⋯

• f

at most countable size, and the distribution function 𝑞 = 𝑞$, we introduce the moments

• f

𝑌, which are numbers defined as follows: 𝜈! = 𝑙th moment of 𝑌 = 𝐹 𝑌! = ∑+,*

• . 𝑦+

!𝑞(𝑦+),

provided the sum

• converges. Here

𝑞 𝑦+ = 𝑄(𝑌 = 𝑦+). ?

𝜈! = ⋯

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=

𝜈$ = 1

?

𝜈% = ⋯

?

𝜈& = ⋯

𝑙th moment

• f

𝑌 𝜈" = 𝐹 𝑌" = *

#$% &'

𝑦#

"𝑞(𝑦#)

𝜈! = 𝐹 1 = *

#$% &'

𝑞(𝑦#) 𝜈% = 𝐹 𝑌 = *

#$% &'

𝑦#𝑞(𝑦#) 𝜈( = 𝐹 𝑌( = *

#$% &'

𝑦#

(𝑞(𝑦#)

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Expected Value & Variance Expe pecte ted Value ue Va Variance

𝜈 = ⋯ 𝜏& = ⋯

𝑙th moment

• f

𝑌 𝜈" = 𝐹 𝑌" = *

#$% &'

𝑦#

"𝑞(𝑦#)

𝜈% = 𝐹 𝑌 = *

#$% &'

𝑦#𝑞(𝑦#) 𝜈( = 𝐹 𝑌( = *

#$% &'

𝑦#

(𝑞(𝑦#)

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Expected Value & Variance Expe pecte ted Value ue Va Variance

𝜈 = 𝜈% 𝜏& = 𝜈& − 𝜈%

& 𝑙th moment

• f

𝑌 𝜈" = 𝐹 𝑌" = *

#$% &'

𝑦#

"𝑞(𝑦#)

𝜈% = 𝐹 𝑌 = *

#$% &'

𝑦#𝑞(𝑦#) 𝜈( = 𝐹 𝑌( = *

#$% &'

𝑦#

(𝑞(𝑦#)

𝜈 = 𝜈%

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Moment Generating Functions

§ We introduce a new variable 𝑢, and define a function 𝑕(𝑢) as follows: 𝑕 𝑢 = 𝐹 𝑓!" = ∑#$%

&' 𝑓!(!𝑞(𝑦#).

§ We call 𝑕(𝑢) the moment generating function for 𝑌, and think

• f

it as a convenient bookkeeping device for describing the moments

• f

𝑌.

=

𝑕 𝑢 = 𝐹 𝑓)* = 𝐹 *

"$! &' 𝑌"𝑢"

𝑙! = *

"$! &' 𝐹(𝑌")𝑢"

𝑙! = *

"$! &' 𝜈"𝑢"

𝑙!

Ex Expect cted val alue 𝑭 𝝔(𝒀) &

"∈$

𝜚(𝑦)𝑛(𝑦)

Taylor Expansion

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Moment Generating Functions

§ If we differentiate 𝑕(𝑢) 𝑜 times and then set 𝑢 = 0, we get 𝜈".

=

𝑕 𝑢 = 𝐹 𝑓)* = *

#$% &'

𝑓)+!𝑞(𝑦#) = *

"$! &' 𝜈"𝑢"

𝑙! 𝑒, 𝑒𝑢, 𝑕 𝑢 = 𝑒, 𝑒𝑢, *

"$! &' 𝜈"𝑢"

𝑙! = *

"$, &' 𝑙! 𝜈"𝑢"-,

𝑙! 𝑙 − 𝑜 ! = *

"$, &' 𝜈"𝑢"-,

𝑙 − 𝑜 ! 𝑒, 𝑒𝑢, 𝑕 𝑢 |)$! = *

"$, &' 𝜈"𝑢"-,

𝑙 − 𝑜 ! |)$! = 𝜈,

=

𝑕 , 0 = 𝑒, 𝑒𝑢, 𝑕 𝑢 |)$! = 𝜈,

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Uniform Distribution

range: 1, 2, 3, ⋯ , 𝑜 distribution function: 𝑞* 𝑘 = %

,

Ra Random vari riable le

𝑕 𝑢 = *

#$% , 1

𝑜 𝑓)# = 1 𝑜 𝑓) + 𝑓() + 𝑓.) + ⋯ + 𝑓,) =

/"(/#"-%) ,(/"-%) .

Generating funct ction

• n

=

𝑕 𝑢 = 𝐹 𝑓%& = &

'() *+

𝑓%"!𝑞(𝑦') = &

,(- *+ 𝜈,𝑢,

𝑙!

=

𝑕 , 0 = 𝑒, 𝑒𝑢, 𝑕 𝑢 |)$! = 𝜈,

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Uniform Distribution

range: 1, 2, 3, ⋯ , 𝑜 distribution function: 𝑞* 𝑘 =

% ,

Ra Random vari riable le

𝑕 𝑢 = ∑#$%

, % , 𝑓)# = /"(/#"-%) ,(/"-%) .

Generating funct ction

• n

𝜈% = 𝑕2 0 =

% , 1 + 2 + 3 + ⋯ + 𝑜 = ,&% ( .

𝜈( = 𝑕22 0 =

% , 1 + 4 + 9 + ⋯ + 𝑜( = (,&%)((,&%) 3

.

Mom Moments

=

𝑕 𝑢 = 𝐹 𝑓%& = &

'() *+

𝑓%"!𝑞(𝑦') = &

,(- *+ 𝜈,𝑢,

𝑙!

=

𝑕 , 0 = 𝑒, 𝑒𝑢, 𝑕 𝑢 |)$! = 𝜈,

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Uniform Distribution

range: 1, 2, 3, ⋯ , 𝑜 distribution function: 𝑞* 𝑘 =

% ,

Ra Random vari riable le

𝑕 𝑢 = ∑#$%

, % , 𝑓)# = /"(/#"-%) ,(/"-%) .

Generating funct ction

• n

𝜈% = 𝑕2 0 =

% , 1 + 2 + 3 + ⋯ + 𝑜 = ,&% ( .

𝜈( = 𝑕22 0 =

% , 1 + 4 + 9 + ⋯ + 𝑜( = (,&%)((,&%) 3

.

Mom Moments

𝜈 = 𝜈% = 𝑜 + 1 2 . 𝜏( = 𝜈( − 𝜈%

( = ,$-% %( .

Expect cted value & variance ce 𝜈 = 𝜈* 𝜏( = 𝜈( − 𝜈*

(

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Binomial Distribution

range: 0, 1, 2, 3, ⋯ , 𝑜 distribution function: 𝑞* 𝑘 =

, # 𝑞#𝑟,-#

Ra Random vari riable le

𝑕 𝑢 = *

#$% ,

𝑓)# 𝑜 𝑘 𝑞#𝑟,-# = ∑#$%

, , # (𝑞𝑓))#𝑟,-# = (𝑞𝑓) + 𝑟),.

Generating funct ction

• n

=

𝑕 𝑢 = 𝐹 𝑓%& = &

'() *+

𝑓%"!𝑞(𝑦') = &

,(- *+ 𝜈,𝑢,

𝑙!

=

𝑕 , 0 = 𝑒, 𝑒𝑢, 𝑕 𝑢 |)$! = 𝜈,

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Binomial Distribution

range: 0, 1, 2, 3, ⋯ , 𝑜 distribution function: 𝑞* 𝑘 =

, # 𝑞#𝑟,-#

Ra Random vari riable le

𝑕 𝑢 = ∑#$%

,

𝑓)#

, # 𝑞#𝑟,-# = (𝑞𝑓) + 𝑟),.

Generating funct ction

• n

𝜈% = 𝑕2 0 = 𝑜(𝑞𝑓) + 𝑟),-%𝑞𝑓)|)$! = 𝑜𝑞. 𝜈( = 𝑕22 0 = 𝑜 𝑜 − 1 𝑞( + 𝑜𝑞.

Mom Moments

=

𝑕 𝑢 = 𝐹 𝑓%& = &

'() *+

𝑓%"!𝑞(𝑦') = &

,(- *+ 𝜈,𝑢,

𝑙!

=

𝑕 , 0 = 𝑒, 𝑒𝑢, 𝑕 𝑢 |)$! = 𝜈,

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Binomial Distribution

range: 0, 1, 2, 3, ⋯ , 𝑜 distribution function: 𝑞* 𝑘 =

, # 𝑞#𝑟,-#

Ra Random vari riable le

𝑕 𝑢 = ∑#$%

,

𝑓)#

, # 𝑞#𝑟,-# = (𝑞𝑓) + 𝑟),.

Generating funct ction

• n

𝜈% = 𝑕2 0 = 𝑜𝑞. 𝜈( = 𝑕22 0 = 𝑜 𝑜 − 1 𝑞( + 𝑜𝑞.

Mom Moments

𝜈 = 𝜈% = 𝑜𝑞. 𝜏( = 𝜈( − 𝜈%

( = 𝑜𝑞(1 − 𝑞).

Expect cted value & variance ce 𝜈 = 𝜈* 𝜏( = 𝜈( − 𝜈*

(

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Geometric Distribution

range: 1, 2, 3, ⋯ , 𝑜 distribution function: 𝑞* 𝑘 = 𝑟#-%𝑞

Ra Random vari riable le

𝑕 𝑢 = ∑#$%

,

𝑓)#𝑟#-%𝑞 =

4/" %-5/".

Generating funct ction

• n

𝜈% = 𝑕2 0 =

4/" (%-5/")$ |)$! = % 4.

𝜈( = 𝑕22 0 =

4/"&45/$" (%-5/")% |)$! = %&5 4$ .

Mom Moments

=

𝑕 𝑢 = 𝐹 𝑓%& = &

'() *+

𝑓%"!𝑞(𝑦') = &

,(- *+ 𝜈,𝑢,

𝑙!

=

𝑕 , 0 = 𝑒, 𝑒𝑢, 𝑕 𝑢 |)$! = 𝜈, 𝜈 = 𝜈% = 1 𝑞 . 𝜏( = 𝜈( − 𝜈%

( = 5 4$.

Expect cted value & variance ce 𝜈 = 𝜈* 𝜏( = 𝜈( − 𝜈*

(

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Poisson Distribution

range: 0, 1, 2, 3, ⋯ , 𝑜 distribution function: 𝑞* 𝑘 = 𝑓-6 6!

#!

Ra Random vari riable le

𝑕 𝑢 = ∑#$%

,

𝑓)#𝑓-6 6!

#! = 𝑓-6 ∑#$% , (6/")! #!

= 𝑓6(/"-%).

Generating funct ction

• n

𝜈% = 𝑕2 0 = 𝑓6(/"-%)𝜇𝑓)|)$! = 𝜇. 𝜈( = 𝑕22 0 = 𝑓6(/"-%) 𝜇(𝑓() + 𝜇𝑓) |)$! = 𝜇( + 𝜇.

Mom Moments

=

𝑕 𝑢 = 𝐹 𝑓%& = &

'() *+

𝑓%"!𝑞(𝑦') = &

,(- *+ 𝜈,𝑢,

𝑙!

=

𝑕 , 0 = 𝑒, 𝑒𝑢, 𝑕 𝑢 |)$! = 𝜈, 𝜈 = 𝜈% = 𝜇. 𝜏( = 𝜈( − 𝜈%

( = 𝜇.

Expect cted value & variance ce

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Bin Binomia ial 𝐹 𝑌 = 𝑜𝑞, 𝑊 𝑌 = 𝑜𝑞𝑟 𝐹 𝑌 = *

/,

𝑊 𝑌 = *#/

/!

𝐹 𝑌 = 𝜇, 𝑊 𝑌 = 𝜇 Ge Geometric Po Poisson 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞!𝑟"#! 𝑄 𝑈 = 𝑜 = 𝑟"#*𝑞 𝑄 𝑌 = 𝑙 = 𝜇! 𝑙! 𝑓#0 𝐹 𝑌 = "-*

( ,

𝑊 𝑌 = "!#*

*(

Un Unifor

• rm

𝑄 𝑌 = 𝑙 = 1 𝑜

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Bin Binomia ial 𝑕 𝑢 = (𝑞𝑓1 + 𝑟)" 𝑕 𝑢 = 𝑞𝑓1 1 − 𝑟𝑓1 𝑕 𝑢 = 𝑓0(3.#*) Ge Geometric Po Poisson 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞!𝑟"#! 𝑄 𝑈 = 𝑜 = 𝑟"#*𝑞 𝑄 𝑌 = 𝑙 = 𝜇! 𝑙! 𝑓#0 𝑕 𝑢 = 𝑓1(𝑓"1 − 1) 𝑜(𝑓1 − 1) Un Unifor

• rm

𝑄 𝑌 = 𝑙 = 1 𝑜

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𝑕 𝑢 = 𝑓0(3.#*). Po Poisson 𝑄 𝑌 = 𝑙 = 𝜇! 𝑙! 𝑓#0

=

𝑕 , 0 = 𝑒, 𝑒𝑢, 𝑕 𝑢 |)$! = 𝜈, 𝐹 𝑌. = 𝜈. = 𝑒. 𝑒𝑢. 𝑕 𝑢 |)$! = 𝑒. 𝑒𝑢. 𝑓6(/"-%)|)$! = 𝑒 𝑒𝑢 𝑓6(/"-%) 𝜇(𝑓() + 𝜇𝑓) |)$! = 𝑓6(/"-%) 𝜇.𝑓.) + 3𝜇(𝑓() + 𝜇𝑓) |)$! = 𝜇. + 3𝜇( + 𝜇

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Mo Moment Problem

§ Let 𝑌 be a discrete random variable with finite range 𝑦%, 𝑦(, ⋯ , 𝑦, , distribution function 𝑞 and moment generation function 𝑕. Then 𝑕 is uniquely determined by 𝑞, and conversely.

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Ordinary Generating Functions

§ In the special but important case where the 𝑦+ are all no nonne nnegati tive inte ntegers, 𝑦+ = 𝑘, we can rewrite the moment generating function in a simpler way: 𝑕 𝑢 = 𝐹 𝑓1$ = ∑!,5

• . &/1/

!! = ∑+,*

• . 𝑓1%0𝑞(𝑦+) = ∑+,*
• . 𝑓1+𝑞(𝑘).

§ We see that 𝑕 𝑢 is a polynomial in 𝑓1. § If we write 𝑨 = 𝑓1, and define the function ℎ by ℎ 𝑨 = ∑+,*

• . 𝑨+𝑞(𝑘),

then ℎ 𝑨 is a polynomial in 𝑨 containing the same information as 𝑕 𝑢 .

=

𝑕 𝑢 = ℎ(𝑓1)

=

ℎ 𝑨 = 𝑕(ln(𝑨))

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Ordinary Generating Functions

§ Moment generating function: 𝑕 𝑢 = 𝐹 𝑓1$ = ∑!,5

• . &/1/

!! = ∑+,*

• . 𝑓1%0𝑞(𝑦+) = ∑+,*
• . 𝑓1+𝑞(𝑘).

§ Ordinary generating function: ℎ 𝑨 = ∑+,*

• . 𝑨+𝑞(𝑘).

=

𝑕 𝑢 = ℎ(𝑓1)

=

ℎ 𝑨 = 𝑕(ln(𝑨))

=

𝑨 = 𝑓1

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Ordinary Generating Functions

§ Moment generating function: 𝑕 𝑢 = 𝐹 𝑓1$ = ∑!,5

• . &/1/

!! = ∑+,*

• . 𝑓1%0𝑞(𝑦+) = ∑+,*
• . 𝑓1+𝑞(𝑘).

§ Ordinary generating function: ℎ 𝑨 = ∑+,*

• . 𝑨+𝑞(𝑘).

ℎ 1 = 𝑕 0 = 1 ℎ′ 1 = 𝑕′ 0 = 𝜈*

ℎ 1 = *

#$% &'

𝑞(𝑘) = 1 ℎ′ 1 = *

#$% &'

𝑘𝑞(𝑘) ℎ22 1 = *

#$% &'

𝑘 𝑘 − 1 𝑞 𝑘 = *

#$% &'

𝑘(𝑞 𝑘 − *

#$% &'

𝑘𝑞 𝑘

ℎ77 * = 𝑕77 5 − 𝑕7 0 = 𝜈( − 𝜈*

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Ordinary Generating Functions

§ Moment generating function: 𝑕 𝑢 = 𝐹 𝑓1$ = ∑!,5

• . &/1/

!! = ∑+,*

• . 𝑓1%0𝑞(𝑦+) = ∑+,*
• . 𝑓1+𝑞(𝑘).

§ Ordinary generating function: ℎ 𝑨 = ∑+,*

• . 𝑨+𝑞(𝑘).

=

Coefficient

• f

𝑨# in ℎ(𝑨): 𝑞 𝑘 = 1 𝑘! 𝑒# 𝑒𝑨# ℎ 𝑨 |8$! = ℎ # (0) 𝑘!

=

𝑕 1 0 = 𝑒, 𝑒𝑢, 𝑕 𝑢 |%(- = 𝜈,

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moments distribution function

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moments MGF OGF distribution function

𝑕 𝑢 = *

"$! &' 𝜈"𝑢"

𝑙! ℎ 𝑨 = 𝑕(ln(𝑨)) 𝑞 𝑘 = ℎ # (0) 𝑘!

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§ 𝑘 = ⋯ § 𝑞 𝑘 = ⋯ 𝜈5 = 1, 𝜈! = *

( + (/ 8 ,

for 𝑙 ≥ 1.

moments distribution function

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moments MGF OGF distribution function

𝜈! = 1, 𝜈" =

% ( + (& 9 ,

for 𝑙 ≥ 1. 𝑕 𝑢 = *

"$! &' 𝜈"𝑢"

𝑙! = 1 + 1 2 *

"$% &' 𝑢"

𝑙! + 1 4 *

"$% &' (2𝑢)"

𝑙! = 1 4 + 1 2 𝑓) + 1 4 𝑓()

Step 1

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moments MGF OGF distribution function

𝜈! = 1, 𝜈" =

% ( + (& 9 ,

for 𝑙 ≥ 1. ℎ 𝑨 = 𝑕 ln 𝑨 = 1 4 + 1 2 𝑨 + 1 4 𝑨( 𝑕 𝑢 = *

"$! &' 𝜈"𝑢"

𝑙! = 1 + 1 2 *

"$% &' 𝑢"

𝑙! + 1 4 *

"$% &' (2𝑢)"

𝑙! = 1 4 + 1 2 𝑓) + 1 4 𝑓()

Step 2

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moments MGF OGF distribution function

𝜈! = 1, 𝜈" =

% ( + (& 9 ,

for 𝑙 ≥ 1. ℎ 𝑨 = 𝑕 ln 𝑨 = 1 4 + 1 2 𝑨 + 1 4 𝑨( 𝑕 𝑢 = *

"$! &' 𝜈"𝑢"

𝑙! = 1 + 1 2 *

"$% &' 𝑢"

𝑙! + 1 4 *

"$% &' (2𝑢)"

𝑙! = 1 4 + 1 2 𝑓) + 1 4 𝑓()

𝑘 = 0, 1, 2 𝑞 𝑘 = 1 4 , 1 2 , 1 4

S t e p 3

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Linearity

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

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Non-linearity

𝑊 𝑑𝑌 = 𝑑(𝑊(𝑌) 𝑊 𝑌 + 𝑑 = 𝑊(𝑌) 𝑊(𝑏𝑌 + 𝑐) = 𝑏(𝑊(𝑌)

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Properties

§ Both the moment generating function 𝑕 and the

• rdinary

generating function ℎ have many properties useful in the study

• f

random variables,

• f

which we can consider

• nly

a few here. 𝑍 = 𝑌 + 𝑏 𝑕9 𝑢 = 𝐹 𝑓19 = 𝐹 𝑓1 $-: = 𝐹 𝑓1:𝑓1$ = 𝑓1:𝐹 𝑓1$ = 𝑓1:𝑕$(𝑢)

=

𝑕 𝑢 = 𝐹 𝑓1$ 𝑍 = 𝑐𝑌 𝑕9 𝑢 = 𝐹 𝑓19 = 𝐹 𝑓1;$ = 𝑕$(𝑐𝑢)

XC 2020

Non-linearity

𝑕KLM 𝑢 = 𝑓NM𝑕K(𝑢) 𝑕OK 𝑢 = 𝑕K(𝑐𝑢)

𝑕%#&

)

𝑢 = 𝑓#&1/)𝑕$(𝑢 𝜏) 𝑍 = 𝑦 − 𝜈 𝜏

XC 2020

Sum of independent random variables discrete & continuous

(𝑔 ∗ 𝑕) 𝑨 = L

#.

• .

𝑔 𝑨 − 𝑧 𝑕 𝑧 𝑒𝑧 𝑛< 𝑘 = P

!

𝑛*(𝑙)𝑛((𝑘 − 𝑙)

co conv nvolu lution

XC 2020

Sum of Independent Random Variables

𝑌 𝑍 𝑎 = 𝑌 + 𝑍 𝑞$ random variable

distribution function (convolution)

𝑞9 𝑞= 𝑘 = P

!

𝑞$(𝑙)𝑞9(𝑘 − 𝑙) MGF 𝑕$(𝑢) 𝑕9(𝑢) 𝑕> 𝑢 = ⋯

=

𝑕 𝑢 = 𝐹 𝑓1$ 𝑕> 𝑢 = 𝐹 𝑓1 $-9 = 𝐹 𝑓1$𝑓19 = 𝐹 𝑓1$ 𝐹 𝑓19 = 𝑕$ 𝑢 𝑕9(𝑢)

XC 2020

𝑌 𝑍 𝑎 = 𝑌 + 𝑍 𝑞$ random variable distribution function 𝑞9 𝑞= 𝑘 = P

!

𝑞$(𝑙)𝑞9(𝑘 − 𝑙) MGF 𝑕$(𝑢) 𝑕9(𝑢) 𝑕> 𝑢 = 𝑕$ 𝑢 𝑕9(𝑢) OGF ℎ$(𝑨) ℎ9(𝑨) ℎ> 𝑨 = ℎ$ 𝑨 ℎ9(𝑨)

=

ℎ 𝑨 = 𝑕(ln(𝑨))

!

independent

XC 2020

Binomial Distribution

range: 0, 1, 2, 3, ⋯ , 𝑜 distribution function: 𝑞* 𝑘 = 𝑞: 𝑘 =

, # 𝑞#𝑟,-#

Ra Random vari riable le

𝑕* 𝑢 = 𝑕: 𝑢 = (𝑞𝑓) + 𝑟),

Mom Moment generating funct ction

• n

=

𝑕> 𝑢 = 𝑕$ 𝑢 𝑕9(𝑢)

=

ℎ> 𝑨 = ℎ$ 𝑨 ℎ9(𝑨)

ℎ* 𝑨 = ℎ: 𝑨 = (𝑞𝑨 + 𝑟),

Ordinary generating funct ction

• n

XC 2020

Binomial Distribution

range: 0, 1, 2, 3, ⋯ , 𝑜 distribution function: 𝑞* 𝑘 = 𝑞: 𝑘 =

, # 𝑞#𝑟,-#

Ra Random vari riable le

𝑕* 𝑢 = 𝑕: 𝑢 = (𝑞𝑓) + 𝑟), 𝑕; 𝑢 = 𝑕* 𝑢 𝑕: 𝑢 = (𝑞𝑓) + 𝑟)(,

Mom Moment generating funct ction

• n

=

𝑕> 𝑢 = 𝑕$ 𝑢 𝑕9(𝑢)

=

ℎ> 𝑨 = ℎ$ 𝑨 ℎ9(𝑨)

ℎ* 𝑨 = ℎ: 𝑨 = (𝑞𝑨 + 𝑟), ℎ; 𝑨 = ℎ* 𝑨 ℎ: 𝑨 = (𝑞𝑨 + 𝑟)(,

Ordinary generating funct ction

• n

XC 2020

Binomial Distribution

𝑕* 𝑢 = 𝑕: 𝑢 = (𝑞𝑓) + 𝑟), 𝑕; 𝑢 = 𝑕* 𝑢 𝑕: 𝑢 = (𝑞𝑓) + 𝑟)(,

Mom Moment generating funct ction

• n

ℎ* 𝑨 = ℎ: 𝑨 = (𝑞𝑨 + 𝑟), ℎ; 𝑨 = ℎ* 𝑨 ℎ: 𝑨 = (𝑞𝑨 + 𝑟)(,

Ordinary generating funct ction

• n

=

ℎ 𝑨 = *

#$% &'

𝑨#𝑞(𝑘) .

ℎ 𝑨 = (𝑞𝑨 + 𝑟)("= P

+,5 ("

2𝑜 𝑘 (𝑞𝑨)+𝑟("#+

range: 0, 1, 2, 3, ⋯ , 𝑜 distribution function: 𝑞* 𝑘 = 𝑞: 𝑘 =

, # 𝑞#𝑟,-#

range: 0, 1, 2, 3, ⋯ , 2𝑜 distribution function: 𝑞; 𝑘 =

(, #

𝑞#𝑟(,-#

Ra Random vari riable le

XC 2020

=

ℎ 𝑨 = P

+,*

• .

𝑨+𝑞(𝑘) . 𝑞 𝑘 = 1 𝑘! 𝑒+ 𝑒𝑨+ ℎ 𝑨 |=,5 = ℎ + (0) 𝑘!

OGF distribution function Taylor Expansion Ready-made polynomials

XC 2020

GENERATING FUNCTIONS

continuous density

XC 2020

Moments

§ If 𝑌 is a continuous random variable defined

• n

the probability space Ω, with density function 𝑔

",

then we define the 𝑜th moment by the formula 𝜈) = 𝐹 𝑌) = ∫

*' &' 𝑦)𝑔 " 𝑦 𝑒𝑦,

provided the integral ∫

*' &' |𝑦|)𝑔 " 𝑦 𝑒𝑦,

is finite. § Then just as the discrete case, we see that

𝜈 = 𝜈* 𝜏( = 𝜈( − 𝜈*

(

1 = 𝜈5

XC 2020

Moment Generating Functions

§ We introduce a new variable 𝑢, and define a function 𝑕(𝑢) as follows: 𝑕 𝑢 = 𝐹 𝑓1$ = ∫

#.

• .𝑓1%𝑔

$ 𝑦 𝑒𝑦.

§ We call 𝑕(𝑢) the moment generating function for 𝑌.

=

𝑕 𝑢 = 𝐹 𝑓)* = 𝐹 *

"$! &' 𝑌"𝑢"

𝑙! = *

"$! &' 𝐹(𝑌")𝑢"

𝑙! = *

"$! &' 𝜈"𝑢"

𝑙!

provided this series converges

=

𝑕 , 0 = 𝑒, 𝑒𝑢, 𝑕 𝑢 |)$! = 𝜈,

XC 2020

Uniform Distribution

range: 0 ≤ 𝑦 ≤ 1 density function: 𝑔

* 𝑘 = 1

Ra Random vari riable le

𝑕 𝑢 = R

! %

𝑓)+𝑒𝑦 = 𝑓) − 1 𝑢

Generating funct ction

• n

=

𝜈, = 𝐹 𝑌, = R

• '

&'

𝑦,𝑔

* 𝑦 𝑒𝑦

=

𝑕 𝑢 = 𝐹 𝑓)* = R

• '

&'

𝑓)+𝑔

* 𝑦 𝑒𝑦

𝜈, = R

! %

𝑦,𝑒𝑦 = 1 𝑜 + 1

Mom Moment

XC 2020

Uniform Distribution

range: 0 ≤ 𝑦 ≤ 1 density function: 𝑔

* 𝑦 = 1

Ra Random vari riable le

𝑕 𝑢 = R

! %

𝑓)+𝑒𝑦 = 𝑓) − 1 𝑢

Generating funct ction

• n

𝜈, = R

! %

𝑦,𝑒𝑦 = 1 𝑜 + 1

Mom Moment

=

𝑕 , 0 = 𝑒, 𝑒𝑢, 𝑕 𝑢 |)$! = 𝜈, 𝜈! = 𝑕 0 = 1 𝜈% = 𝑕2 0 = 1 2 𝜈( = 𝑕22 0 = 1 3 𝜈 = 𝜈% = 1 2 𝜏( = 𝜈( − 𝜈%

( = 1

12

XC 2020

Exponential Distribution

range: 𝑦 ≥ 0 density function: 𝑔

* 𝑦 = 𝜇𝑓-6+

Ra Random vari riable le

𝑕 𝑢 = R

! &'

𝑓)+𝜇𝑓-6+𝑒𝑦 = 𝜇 𝜇 − 𝑢

Generating funct ction

• n

=

𝜈, = 𝐹 𝑌, = R

• '

&'

𝑦,𝑔

* 𝑦 𝑒𝑦

=

𝑕 𝑢 = 𝐹 𝑓)* = R

• '

&'

𝑓)+𝑔

* 𝑦 𝑒𝑦

𝜈, = R

! &'

𝑦,𝜇𝑓-6+𝑒𝑦 = 𝑜! 𝜇,

Mom Moment

XC 2020

Exponential Distribution

range: 𝑦 ≥ 0 density function: 𝑔

* 𝑦 = 𝜇𝑓-6+

Ra Random vari riable le

𝑕 𝑢 = R

! &'

𝑓)+𝜇𝑓-6+𝑒𝑦 = 𝜇 𝜇 − 𝑢

Generating funct ction

• n

𝜈, = R

! &'

𝑦,𝜇𝑓-6+𝑒𝑦 = 𝜇𝑜! (𝜇 − 𝑢),&% |)$! = 𝑜! 𝜇,

Mom Moment

=

𝑕 , 0 = 𝑒, 𝑒𝑢, 𝑕 𝑢 |)$! = 𝜈,

XC 2020

Normal Distribution (standard)

range: −∞ ≤ 𝑦 ≤ +∞ density function: 𝑔

* 𝑦 = % (< 𝑓-+$/(

Ra Random vari riable le

𝑕 𝑢 = 1 2𝜌 R

• '

&'

𝑓)+𝑓-+$/(𝑒𝑦 = 𝑓)$/(

Generating funct ction

• n

=

𝜈, = 𝐹 𝑌, = R

• '

&'

𝑦,𝑔

* 𝑦 𝑒𝑦

=

𝑕 𝑢 = 𝐹 𝑓)* = R

• '

&'

𝑓)+𝑔

* 𝑦 𝑒𝑦

§ 𝜈, =

% (< ∫

• '

&' 𝑦,𝑓-+$/(𝑒𝑦 = (> ! ('>! if

𝑜 = 2𝑛 § 𝜈, = 0 if 𝑜 = 2𝑛 + 1

Mom Moment

XC 2020

Normal Distribution (general)

range: −∞ ≤ 𝑦 ≤ +∞ density function: 𝑔

* 𝑦 = % (<? 𝑓-(+-@)$/(?$

Ra Random vari riable le

𝑕 𝑢 = 1 2𝜌𝜏 R

• '

&'

𝑓)+𝑓-(+-@)$/(?$𝑒𝑦 = 𝑓@)&?$)$/(

Generating funct ction

• n

=

𝑕 𝑢 = 𝑓1!/(

=

𝑕*&A 𝑢 = 𝑓)A𝑕*(𝑢) 𝑕B* 𝑢 = 𝑕*(𝑐𝑢) confluent hypergeometric function

Mom Moment

XC 2020

Normal Distribution (sum)

range: −∞ ≤ 𝑦, 𝑧 ≤ +∞ distribution function: 𝑔

* 𝑦 =

1 2𝜌𝜏% 𝑓-(+-@()$/(?($ 𝑔

: 𝑦 =

1 2𝜌𝜏( 𝑓-(+-@$)$/(?$$

Ra Random vari riable le

𝑕* 𝑢 = 𝑓@()&?($)$/( 𝑕: 𝑢 = 𝑓@$)&?$$)$/( 𝑕; 𝑢 = 𝑓(@(&@$))&(?($&?$$))$/(

Mom Moment generating funct ction

• n

=

𝑕> 𝑢 = 𝑕$ 𝑢 𝑕9(𝑢) 𝐹 𝑎 = 𝜈* + 𝜈( 𝑊 𝑎 = 𝜏*( + 𝜏((

XC 2020

Mo Moment Problem

§ If 𝑌 is a bounded random variable, then the moment generating function 𝑕$ 𝑢 determines the density function 𝑔

$ 𝑦

uniquely.

XC 2020