MATH 20: PROBABILITY Variance of Discrete Random Variables - - PowerPoint PPT Presentation

MATH 20: PROBABILITY

Variance

• f

Discrete Random Variables Xingru Chen xingru.chen.gr@dartmouth.edu

XC 2020

Important Distributions

𝑛 𝜕 = 1 𝑜

Discrete Uniform Distribution

𝑄 𝑈 = 𝑜 = 𝑟!"#𝑞

Geometric Distribution

𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞$𝑟!"$

Binomial Distribution

ℎ 𝑂, 𝑙, 𝑜, 𝑦 =

$ % &"$ !"% & !

Hypergeometric Distribution

𝑣 𝑦, 𝑙, 𝑞 = 𝑦 − 1 𝑙 − 1 𝑞$𝑟%"$

Negative Binomial Distribution

𝑄 𝑌 = 𝑙 = 𝜇$ 𝑙! 𝑓"'

Poisson Distribution

XC 2020

Bin Binomia ial 𝐹 𝑌 = 𝑜𝑞 𝐹 𝑌 = 1 𝑞 𝐹 𝑌 = 𝜇 𝐹 𝑌 = 𝑙 𝑟 𝑞 Ge Geometric Po Poisson Ne Negative binomi

• mial

𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞!𝑟"#! 𝑄 𝑈 = 𝑜 = 𝑟"#$𝑞 𝑄 𝑌 = 𝑙 = 𝜇! 𝑙! 𝑓#% 𝑣 𝑦, 𝑙, 𝑞 = 𝑦 − 1 𝑙 − 1 𝑞!𝑟&#! 𝐹 𝑌 = 𝑜 𝑙 𝑂 Hy Hyper ergeo eomet etric ic ℎ 𝑂, 𝑙, 𝑜, 𝑦 =

! & '#! "#& ' "

XC 2020

How Much Does a Hershey Kiss Weight?

§ A single standard Hershey's Kiss weighs 0.16

• unces.

XC 2020

?

Ho How t to m mea easu sure t the q qualit ity

• f

prod

• duct

cts? Same 𝜈, different 𝜏.

XC 2020

Home Made Versus Factory Made

XC 2020

Variance of Discrete Random Variables

§ Let 𝑌 be a numerically-valued random variable with expected value 𝜈 = 𝐹(𝑌). Then the variance

• f

𝑌, denoted by 𝑊(𝑌), is 𝑊 𝑌 = 𝐹((𝑌 − 𝜈)().

Ex Expected value 𝑭(𝒀) :

%∈)

𝑦𝑛(𝑦) Ex Expected value 𝑭 𝝔(𝒀) :

%∈)

𝜚(𝑦)𝑛(𝑦) Va Variance 𝑊 𝑌 :

%∈)

(𝑦 − 𝜈)*𝑛(𝑦)

XC 2020

Variance of Discrete Random Variables

§ Let 𝑌 be a numerically-valued random variable with expected value 𝜈 = 𝐹(𝑌). Then the variance

• f

𝑌, denoted by 𝑊(𝑌), is 𝑊 𝑌 = 𝐹((𝑌 − 𝜈)(). § Standard deviation

• f

𝑌, denoted by 𝐸(𝑌), is 𝐸 𝑌 = 𝑊(𝑌). § We

• ften

write 𝜏 for 𝐸(𝑌) and 𝜏( for 𝑊 𝑌 .

Va Variance 𝑊 𝑌 :

%∈)

(𝑦 − 𝜈)*𝑛(𝑦)

XC 2020

Variance of Discrete Random Variables

§ Let 𝑌 be a numerically-valued random variable with expected value 𝜈 = 𝐹(𝑌). Then the variance

• f

𝑌, denoted by 𝑊(𝑌), is 𝑊 𝑌 = 𝐹((𝑌 − 𝜈)(). § If 𝑌 is any random variable with 𝜈 = 𝐹(𝑌), then 𝑊 𝑌 = 𝐹 𝑌( − 𝜈(.

Va Variance 𝑊 𝑌 :

%∈)

(𝑦 − 𝜈)*𝑛(𝑦) :

%∈)

𝑦*𝑛(𝑦) − :

%∈)

𝑦𝑛(𝑦)

XC 2020

Let 𝑌 be a numerically-valued random variable with expected value 𝜈 = 𝐹(𝑌). 𝑊 𝑌 = :

%∈)

(𝑦 − 𝜈)*𝑛(𝑦) = :

%∈)

(𝑦* − 2𝜈𝑦 + 𝜈*)𝑛(𝑦) = :

%∈)

𝑦*𝑛(𝑦) − 2𝜈 :

%∈)

𝑦𝑛 𝑦 + 𝜈* :

%∈)

𝜈*𝑛 𝑦

Proof

𝑊 𝑌 = $

!∈#

(𝑦 − 𝜈)$𝑛(𝑦)

:

%∈)

𝑦𝑛 𝑦 = 𝜈

$

!∈#

𝑛 𝑦 = 1

𝑊 𝑌 = :

%∈)

𝑦*𝑛(𝑦) − 2𝜈 :

%∈)

𝑦𝑛 𝑦 + 𝜈* :

%∈)

𝜈*𝑛 𝑦 = 𝐹 𝑌* − 2𝜈* + 𝜈* = 𝐹 𝑌* − 𝜈* :

%∈)

𝑦*𝑛(𝑦) = 𝐹(𝑌*)

XC 2020

Example 1

Tos

• ss

a coi coin head

• r

tail 1 or 𝑛 𝑦 = 1 2 𝜈 = 𝐹 𝑌 = 1 2 𝜏( = 𝑊 𝑌 = ⋯

XC 2020

Example 1

Tos

• ss

a coi coin head

• r

tail 1 or 𝑛 𝑦 = 1 2 𝜈 = 𝐹 𝑌 = 1 2 𝜏( = 𝑊 𝑌 = 1 4

XC 2020

Example 2

Rol

• ll

a dice ce 1, 2, 3, 4, 5,

• r

6 𝑛 𝑦 = 1 6 𝜈 = 𝐹 𝑌 = 7 2 𝜏( = 𝑊 𝑌 = ⋯

XC 2020

Example 2

Rol

• ll

a dice ce 1, 2, 3, 4, 5,

• r

6 𝑛 𝑦 = 1 6 𝜈 = 𝐹 𝑌 = 7 2 𝜏( = 𝑊 𝑌 = 91 6 − 49 4 = 35 12

XC 2020

Quiz 9

What is the expected number

• f

dice tosses needed to get two consecutive six's?

Number

• f

tosses 2, 3, 4, … 𝐹 𝑌 = 𝐹 𝑌 1 𝑄 1 + 𝐹 𝑌 2 + 𝐹 𝑌 3 𝑄 3 + 𝐹 𝑌 4 𝑄 4 + 𝐹 𝑌 5 𝑄 5 + 𝐹 𝑌 6 𝑄(6) 𝐹 𝑌 = 5 6 𝐹 𝑌 1 + 1 6 𝐹 𝑌 6 = 5 6 1 + 𝐹 𝑌 + 1 6 (5 6 𝐹 𝑌 61 + 1 6 𝐹(𝑌|66)) 𝐹 𝑌 = 5 6 1 + 𝐹 𝑌 + 1 6 [5 6 2 + 𝐹 𝑌 + 2 6]

XC 2020

Is the distribution function a must for calculating expectation?

𝑭 𝒀

𝑭 𝒀 𝑮𝟐 𝑸(𝑮𝟐) 𝑭 𝒀 𝑮𝟕 𝑸(𝑮𝟕) 𝑭 𝒀 𝑮𝟔 𝑸(𝑮𝟔) 𝑭 𝒀 𝑮𝟑 𝑸(𝑮𝟑) 𝑭 𝒀 𝑮𝟓 𝑸(𝑮𝟓) 𝑭 𝒀 𝑮𝟒 𝑸(𝑮𝟒)

XC 2020

Example 3

Consider the general Bernoulli trial

• process. As

usual, we let 𝑌 = 1 if the

• utcome

is a success and 0 if it is a failure. Expect cted value 𝑭(𝒀) M

&∈0

𝑦𝑛(𝑦) = 1×𝑞 + 0× 1 − 𝑞 = 𝑞 Ber Bernoulli t tria ial 𝑛 𝑦 = O 𝑞, 𝑌 = 1 1 − 𝑞, 𝑌 = 0 Variance ce 𝑊 𝑌 𝐹 𝑌( − 𝜈( = ⋯

XC 2020

Example 3

Consider the general Bernoulli trial

• process. As

usual, we let 𝑌 = 1 if the

• utcome

is a success and 0 if it is a failure. Expect cted value 𝑭(𝒀) M

&∈0

𝑦𝑛(𝑦) = 1×𝑞 + 0× 1 − 𝑞 = 𝑞 Ber Bernoulli t tria ial 𝑛 𝑦 = O 𝑞, 𝑌 = 1 1 − 𝑞, 𝑌 = 0 Variance ce 𝑊 𝑌 𝐹 𝑌( − 𝜈( = 𝑞 − 𝑞(

XC 2020

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

1,

𝑊 𝑌 = $#1

1%

𝐹 𝑌 = 𝜇, 𝑊 𝑌 = 𝜇 𝐹 𝑌 = 𝑙 2

1,

𝑊 𝑌 = 𝑙 2

1%

Ge Geometric Po Poisson Ne Negative binomi

• mial

𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞!𝑟"#! 𝑄 𝑈 = 𝑜 = 𝑟"#$𝑞 𝑄 𝑌 = 𝑙 = 𝜇! 𝑙! 𝑓#% 𝑣 𝑦, 𝑙, 𝑞 = 𝑦 − 1 𝑙 − 1 𝑞!𝑟&#! 𝐹 𝑌 = 𝑜 !

',

𝑊 𝑌 = "! '#! ('#")

'%('#$)

Hy Hyper ergeo eomet etric ic ℎ 𝑂, 𝑙, 𝑜, 𝑦 =

! & '#! "#& ' "

XC 2020

Binomial Distribution and Poisson Distribution

Po Poisson Distribution 𝑄 𝑌 = 𝑙 = 𝜇$ 𝑙! 𝑓"' 𝐹 𝑌 = 𝜇, 𝑊 𝑌 = 𝜇 Bi Binomial Dist stribution 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞$𝑟!"$ 𝐹 𝑌 = 𝑜𝑞, 𝑊 𝑌 = 𝑜𝑞𝑟

=

𝑞 = %5

",

𝑢 = 1, 𝑜 → ∞, 𝑞 → 0

XC 2020

Bin Binomia ial 𝐹 𝑌 = 𝑜𝑞 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞!𝑟"#! Variance ce 𝑊 𝑌 𝐹 𝑌( − 𝜈( = M

&∈0

𝑦(𝑛 𝑦 − 𝜈( M

&∈0

𝑦(𝑛 𝑦 = M

!67 "

𝑙( 𝑜 𝑙 𝑞!𝑟"#! = M

!6$ "

𝑙( 𝑜 𝑙 𝑞!𝑟"#! = M

!6$ "

𝑙( 𝑜! 𝑙! 𝑜 − 𝑙 ! 𝑞!𝑟"#! = M

!6$ "

𝑜𝑞𝑙 (𝑜 − 1)! (𝑙 − 1)! 𝑜 − 𝑙 ! 𝑞!#$𝑟"#! = 𝑜𝑞 M

!6$ "

(𝑙 − 1 + 1) 𝑜 − 1 𝑙 − 1 𝑞!#$𝑟"#! = 𝑜𝑞 M

867 "#$

𝑚 𝑜 − 1 𝑚 𝑞8𝑟"#$#8 + 𝑜𝑞 M

867 "#$ 𝑜 − 1

𝑚 𝑞8𝑟"#$#8

XC 2020

Bin Binomia ial 𝐹 𝑌 = 𝑜𝑞 𝑐 𝑜, 𝑞, 𝑙 = 𝑜 𝑙 𝑞!𝑟"#! Variance ce 𝑊 𝑌 𝐹 𝑌( − 𝜈( = M

&∈0

𝑦(𝑛 𝑦 − 𝜈( § ∑&∈0 𝑦(𝑛 𝑦 = 𝑜 𝑜 − 1 𝑞( ∑86$

"#$ "#( 8#$ 𝑞8#$𝑟"#$#8 + 𝑜𝑞 = 𝑜 𝑜 − 1 𝑞( + 𝑜𝑞

§ 𝜈( = 𝑜(𝑞( § ∑&∈0 𝑦(𝑛 𝑦 − 𝜈( = 𝑜 𝑜 − 1 𝑞( + 𝑜𝑞 − 𝑜(𝑞( = 𝑜𝑞(1 − 𝑞)

XC 2020

𝐹 𝑌 = 𝜇 Po Poisson 𝑄 𝑌 = 𝑙 = 𝜇! 𝑙! 𝑓#% Variance ce 𝑊 𝑌 𝐹 𝑌( − 𝜈( = M

&∈0

𝑦(𝑛 𝑦 − 𝜈( M

&∈0

𝑦(𝑛 𝑦 = M

!67 9:

𝑙( 𝜇! 𝑙! 𝑓#% = M

!6$ 9:

𝑙 𝜇! 𝑙! 𝑓#% = M

!6$ 9:

𝜇𝑙 𝜇!#$ (𝑙 − 1)! 𝑓#% = M

!6$ 9:

𝜇(𝑙 − 1 + 1) 𝜇!#$ (𝑙 − 1)! 𝑓#% = 𝜇 M

867 9:

𝑚 𝜇8 𝑚! 𝑓#% + 𝜇 M

867 9: 𝜇8

𝑚! 𝑓#%

XC 2020

𝐹 𝑌 = 𝜇 Po Poisson 𝑄 𝑌 = 𝑙 = 𝜇! 𝑙! 𝑓#% Variance ce 𝑊 𝑌 𝐹 𝑌( − 𝜈( = M

&∈0

𝑦(𝑛 𝑦 − 𝜈( § ∑&∈0 𝑦(𝑛 𝑦 = 𝜇 ∑867

9: 𝑚 %& 8! 𝑓#% + 𝜇 ∑867 9: %& 8! 𝑓#% = 𝜇( ∑86$ 9: %&'( (8#$)! 𝑓#% + 𝜇 = 𝜇( + 𝜇

§ 𝜈( = 𝜇( § ∑&∈0 𝑦(𝑛 𝑦 − 𝜈( = 𝜇( + 𝜇 − 𝜇( = 𝜇

XC 2020

Linearity

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

XC 2020

Properties of Variance

§ If 𝑌 is any random variable and 𝑑 is any constant, then 𝑊 𝑑𝑌 = 𝑑(𝑊(𝑌), 𝑊 𝑌 + 𝑑 = 𝑊(𝑌). Variance ce 𝑊 𝑌 = 𝐹 𝑌( − 𝜈( = 𝐹 𝑌( − [𝐹 𝑌 ]( 𝑊 𝑑𝑌 = 𝐹 (𝑑𝑌)( − [𝐹 𝑑𝑌 ]( 𝐹(𝑏𝑌 + 𝑐) = 𝑏𝐹(𝑌) + 𝑐

XC 2020

Properties of Variance

§ If 𝑌 is any random variable and 𝑑 is any constant, then 𝑊 𝑑𝑌 = 𝑑(𝑊(𝑌), 𝑊 𝑌 + 𝑑 = 𝑊(𝑌). Variance ce 𝑊 𝑌 = 𝐹 𝑌( − 𝜈( = 𝐹 𝑌( − [𝐹 𝑌 ]( 𝑊 𝑑𝑌 = 𝐹 (𝑑𝑌)( − 𝐹 𝑑𝑌

( = 𝐹 𝑑(𝑌( − 𝑑𝐹 𝑌 (

= 𝑑(𝐹 𝑌( − 𝑑( 𝐹 𝑌

( = 𝑑( 𝐹 𝑌( − 𝐹 𝑌 (

= 𝑑(𝑊 𝑌 𝐹(𝑏𝑌 + 𝑐) = 𝑏𝐹(𝑌) + 𝑐

XC 2020

Properties of Variance

§ If 𝑌 is any random variable and 𝑑 is any constant, then 𝑊 𝑑𝑌 = 𝑑(𝑊(𝑌), 𝑊 𝑌 + 𝑑 = 𝑊(𝑌). Variance ce 𝑊 𝑌 = 𝐹 𝑌 − 𝜈 ( = 𝐹 𝑌 − 𝐹(𝑌) ( 𝑊 𝑌 + 𝑑 = 𝐹 𝑌 + 𝑑 − 𝐹(𝑌 + 𝑑) ( 𝐹(𝑏𝑌 + 𝑐) = 𝑏𝐹(𝑌) + 𝑐

XC 2020

Properties of Variance

§ If 𝑌 is any random variable and 𝑑 is any constant, then 𝑊 𝑑𝑌 = 𝑑(𝑊(𝑌), 𝑊 𝑌 + 𝑑 = 𝑊(𝑌). Variance ce 𝑊 𝑌 = 𝐹 𝑌 − 𝜈 ( = 𝐹 𝑌 − 𝐹(𝑌) ( 𝑊 𝑌 + 𝑑 = 𝐹 𝑌 + 𝑑 − 𝐹(𝑌 + 𝑑) ( = 𝐹 𝑌 + 𝑑 − 𝐹 𝑌 − 𝑑 ( = 𝐹 𝑌 − 𝐹 𝑌

(

= 𝑊(𝑌). 𝐹(𝑏𝑌 + 𝑐) = 𝑏𝐹(𝑌) + 𝑐

XC 2020

Non-linearity

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

XC 2020

When do we need independence?

𝑌 + 𝑍 𝑌𝑍 Ne Need

𝐹(𝑌𝑍) = 𝐹(𝑌)𝐹(𝑍) …

Do Do not need

𝐹(𝑌 + 𝑍) = 𝐹(𝑌) + 𝐹(𝑍) … … 𝑾(𝒀 + 𝒁) = 𝑾(𝒀) + 𝑾(𝒁)

XC 2020

When do we need independence?

𝑌 + 𝑍 𝑌𝑍 Ne Need

𝐹(𝑌𝑍) = 𝐹(𝑌)𝐹(𝑍) 𝑊(𝑌 + 𝑍) = 𝑊(𝑌) + 𝑊(𝑍) …

Do Do not need

𝐹(𝑌 + 𝑍) = 𝐹(𝑌) + 𝐹(𝑍) … …

XC 2020

Let 𝑌 and 𝑍 be random variables with finite expected values. And 𝑌 and 𝑍 are independent random variables.

𝑊 𝑌 + 𝑍 = 𝐹 𝑌 + 𝑍 ( − 𝑏 + 𝑐 ( = 𝐹 𝑌( + 2𝑌𝑍 + 𝑍( − 𝑏 + 𝑐 ( = 𝐹 𝑌( + 2𝐹 𝑌𝑍 + 𝐹 𝑍( − 𝑏( − 2𝑏𝑐 − 𝑐(

Proof

𝑊 𝑌 = 𝐹 𝑌( − 𝜈(

𝐹 𝑍 = 𝑐

𝐹(𝑌𝑍) = 𝐹(𝑌)𝐹(𝑍)

𝑊 𝑌 + 𝑍 = 𝐹 𝑌* + 2𝐹 𝑌 𝐹 𝑍 + 𝐹 𝑍* − 𝑏* − 2𝑏𝑐 − 𝑐* = 𝐹 𝑌* + 2𝑏𝑐 + 𝐹 𝑍* − 𝑏* − 2𝑏𝑐 − 𝑐* = 𝐹 𝑌* − 𝑏* + 𝐹 𝑍* − 𝑐* = 𝑊 𝑌 + 𝑊(𝑍) 𝐹 𝑌 = 𝑏 𝐹 𝑌 + 𝑍 = 𝑏 + 𝑐

XC 2020

Properties of Variance

§ If 𝑌 is any random variable and 𝑑 is any constant, then 𝑊 𝑑𝑌 = 𝑑(𝑊(𝑌), 𝑊 𝑌 + 𝑑 = 𝑊(𝑌). § Let 𝑌 and 𝑍 be two in independent random

• variables. Then

𝑊(𝑌 + 𝑍) = 𝑊(𝑌) + 𝑊(𝑍). § It can be shown that the variance

• f

the sum

• f

any number

• f

mutually independent random variables is the sum

• f

the individual variances.

XC 2020

Properties of Variance

§ Let 𝑌$, 𝑌(, ⋯ , 𝑌" be an independent trials process with 𝐹 𝑌

< = 𝜈 and

𝑊 𝑌

< = 𝜏(.

Let 𝑇" = 𝑌$ + 𝑌( + ⋯ + 𝑌" be the sum, and 𝐵" = =)

" be

the

• average. Then

𝑊 𝑑𝑌 = 𝑑*𝑊(𝑌) independent 𝑊(𝑌 + 𝑍) = 𝑊(𝑌) + 𝑊(𝑍).

?

𝐹 𝑇" = ⋯

?

𝑊 𝑇" = ⋯

XC 2020

Properties of Variance

§ Let 𝑌$, 𝑌(, ⋯ , 𝑌" be an independent trials process with 𝐹 𝑌

< = 𝜈 and

𝑊 𝑌

< = 𝜏(.

Let 𝑇" = 𝑌$ + 𝑌( + ⋯ + 𝑌" be the sum, and 𝐵" = =)

" be

the

• average. Then

𝑊 𝑑𝑌 = 𝑑$𝑊(𝑌) independent 𝑊(𝑌 + 𝑍) = 𝑊(𝑌) + 𝑊(𝑍).

= 𝐹 𝑇" = 𝑜𝜈 = 𝑊 𝑇" = 𝑜𝜏(

?

𝐹 𝐵" = ⋯

?

𝑊 𝐵" = ⋯, 𝐸 𝐵" = ⋯

XC 2020

Properties of Variance

§ Let 𝑌$, 𝑌(, ⋯ , 𝑌" be an independent trials process with 𝐹 𝑌

< = 𝜈 and

𝑊 𝑌

< = 𝜏(.

Let 𝑇" = 𝑌$ + 𝑌( + ⋯ + 𝑌" be the sum, and 𝐵" = =)

" be

the

• average. Then

= 𝐹 𝑇" = 𝑜𝜈 = 𝑊 𝑇" = 𝑜𝜏(

=

𝐹 𝐵" = 𝜈

=

𝑊 𝐵" = >%

" ,

𝐸 𝐵" = >

"

XC 2020

Ber Bernoulli t tria ial 𝑛 𝑦 = O 𝑞, 𝑌 = 1 1 − 𝑞, 𝑌 = 0 𝑭 𝒀 = 𝑞 𝑊 𝑌 = 𝑞 − 𝑞(

Example

Bin Binomia ial n independent Bernoulli trials = 𝐹 𝑇" = 𝑜𝜈 = 𝑊 𝑇" = 𝑜𝜏(

XC 2020

Law of Large Numbers

§ Let 𝑌$, 𝑌(, ⋯ , 𝑌" be an independent trials process with 𝐹 𝑌

< = 𝜈 and

𝑊 𝑌

< = 𝜏(.

§ Let 𝑇" = 𝑌$ + 𝑌( + ⋯ + 𝑌" be the sum, and 𝐵" = =)

" be

the

• average. Then

=

𝐹 𝐵" = 𝜈

=

𝑊 𝐵" = >%

" ,

𝐸 𝐵" = >

"

𝑜 → +∞ 𝑊 𝐵" → ⋯ 𝐸 𝐵" → ⋯

XC 2020