Joint Probability Mass and Density Functions of Discrete and Continuous Random Variables, Study notes of Calculus

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Random Variables and their

Expectation: Part III^ Cyr Emile M’LAN, Ph.D.

[email protected]^ Random Variable and Expectation: Part III

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Introduction

♠^ Text Reference

:^ Introduction to Probability and

Statistics for Engineers and Scientists, Chapter 2. ♠ Reading Assignment

:^ Sections 4.3, 4.7, 4.8, and

4.9, October 20 - October 22 So far we have studied the probability models for asingle random variable. Many problems in probabilityand statistics lead to models involving several randomvariables simultaneously. Thus, in this lecture notes, we discuss probabilitymodels for the joint behavior of several randomvariables. We revisit and reemphasize the notion of independenceand of conditional probability but in the context of two ormore random variables.

Random Variable and Expectation: Part III

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Jointly Distributed Discrete Random Variables^ Definition 4.

Let^ X^ and

Y^ be two discrete random variables associated to a random experiment, each assuming values on the samplespace^

Sand^1

S, respectively.^2

The^ joint probability mass

function,

p(x, y)

,^ is defined for each pair of numbers

(x, y)^ in

S^ =^ S^1

⊗ Sby^2

p(x, y) =

P^ (X^ =

x, Y^ =

y)^.

Let^ A^ be any event in the sample space

S, that is, consisting

of pairs

(x, y)^ with

x^ ∈ S^1

and^ y^ ∈ S. Then the probability^2

that the random pair

(X, Y^ )

lies in A is obtained by summing

the joint probability mass function over pairs

(x, y)^ in

A:

[ P (X, Y

]^ ) ∈ A ∑= (x,y)∈ p(x, yA )^.

Random Variable and Expectation: Part III

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Jointly Distributed Discrete Random Variables^ Theorem 4.

Let^ X^

and^ Y^

be two discrete random variables with joint probability mass function,

p(x, y)

, defined on

S.^ Then, it

must^ p(

x, y)^ must satisfy:1. p(x, y)^ ≥^0

for all^ x

and^ y. ∑2. (x,y) p(x, y∈S ) = 1^.

Definition 4.

Let^ X^

and^ Y^

be two discrete random variables with joint probability mass function,

p(x, y)

, defined on

S.^ The

joint

distribution function,

F^ (x, y)

,^ is defined as

F^ (x, y) =

P^ (X^ ≤

x, Y^ ≤

∑ y) =? x ∑ ?≤x y≤y

?? p(x, y )

Random Variable and Expectation: Part III

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Jointly Distributed Discrete Random Variables^ Example 4.

An large insurance agency services a number of customers who have purchased botha homeowner’s policy and an automobile policy from the agency. For each type ofpolicy, a deductible amount must be specified. For an automobile policy, the choicesare $100 and $250, whereas for a homeowner’s policy, the choices are $0, $100, and$200. Suppose an individual with both types of policy is selected at random from theagency’s files. Let

X^ = the deductible amount on the auto policy and

Y^ = the

deductible amount on the homeowner’s policy. Suppose the joint probability massfunction is

y p(x, y)^

0 100

200 x^100

.20^.

.

.^

.

a).^ Find^ P

(X^ = 100

, Y^ ≥^ 100)

.

b).^ Find the marginal probability mass function of

X^ and^ Y^

and^ P^ (Y^

≥^ 100).

Random Variable and Expectation: Part III

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Jointly Distributed Discrete Random Variables^ Solution

a).^ P^ (X

= 100, Y

≥^ 100) =

p(100,^

100. +^

p(100,^ 200) =

.10 +^ .20 =

.^30. b).^ The marginal probability mass function of

X^ and

Y^ are:

y p(x, y)^

0 100

200

p(x)X^

x^100

.^

.10^.

.

.^ .^

.

p(y)^ Y^

.25^.

.^

Hence, P^ (Y^ ≥

100. =

P^ (Y^ = 100) +

P^ (Y^ = 250) =

.25 +^.

50 =^.^75

Random Variable and Expectation: Part III

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Jointly Distributed Continuous Random Variables^ Theorem 4.

Let^ X^ and

Y^ be two continuous random variables with joint probability density function,

f^ (x, y)

. Then,

f^ (x, y)

must sat-

isfy:1.^ f^ (x, y

)^ ≥^0 for all

x^ and^

y.

∫^ ∞ 2.−∞ ∫^ ∞^ f^ (x, y−∞

)^ dx dy^

= 1^.

Definition 4.

Let^ X^ and

Y^ be two continuous random variables with joint probability density function,

f^ (x, y

).^ The

joint distribution

function,

F^ (x, y)

is defined as F^ (x, y) =

P^ (X^ ≤

x, Y^ ≤

∫ y) = ∫^ yx −∞^ −∞

?? f (x, y

?^ ) dxdy ?

Random Variable and Expectation: Part III

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Jointly Distributed Continuous Random Variables^ Definition 4.

Let^ X^ and

Y^ be two continuous random variables with joint probability density function,

f^ (x, y)

. The^ marginal probability

density functions of

X^ and

Y^ ,^ f(X^

x)^ and^

f(y), respectively,Y^

are given by as

f(x)^ X^

∫^ ∞ =−∞ f^ (x, y)^

dy

f(y)^ Y^

∫^ ∞ =−∞ f^ (x, y)^

dx Random Variable and Expectation: Part III

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Jointly Distributed Continuous Random Variables^ Solution

a).^ We have ∫ ∫^ ∞∞ −∞−∞

f^ (x, y)^ dx dy

∫ = ∫^1165

(^2) (x + y) dx dy ∫ (^6) = 5 ∫^11 x dx dy 00

∫^16 + 50 ∫^12 ydx dy^0

∫ (^6) = 5 1 x dx^ + 0

∫^162 y 50

(^6) dy = 10 (^6) + = 1 15

b).^ We have (^1 P X^ ≤^ , Y^4

) (^1) ≤ 4

∫ (^6) = 5 ∫^1 /^41 / 0

4 2 (x^ +^ y 0

)^ dx dy ∫ (^6) = 5 ∫^1 /^41 / 0

4 x dx dy 0

∫^6 + 5 ∫^1 / 41 /^400 (^2) ydx dy

(^6) = · 20 ∣^ x=1 2 ∣x∣ ∣ 2 /^46 +^20 x==

∣^ y=1 3 ∣y∣· ∣ 3 /^43 =^320 y==

(^1) + = 640 (^7 )

Random Variable and Expectation: Part III

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Jointly Distributed Continuous Random Variables^ Solution

c).^ The marginal density function of

X^ and^

y^ are

f(x)^ X^

∫^ ∞ =−∞ f^ (x, y)^ dy

∫^6 = 5 1 x dy^ + 0

∫^162 y 50 dy

(^6) = x^5

∣ 3 ∣ 6 y∣+ ∣ 53 y=1^ = y==

62 x^ + 5 5

f(y)^ Y^

∫^6 = 5 1 (x^ +^ y 0

2 )^ dx^ =

∫^16 x dx 50

∫^16 + 50 (^2) ydx

6 x = 5 ∣^ x=1 2 ∣∣^ + ∣ 2 x==

662 y= 5 5

(^32) y+ 5

c).^ We have^ P

(^1 ≤^ Y^4

) (^3) ≤ 4

∫^3 / = 4 f(y)^ Y^1 / 4

∫^ dy = (^162 y^5

) (^3) +^ dy 5

37 Random Variable and Expectation: Part III = 80

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Independent Random Variables Definition 4.

Two random variables

X^ and

Y^ are said to be

independent

if for any two sets of real number

A^ and^

B

P^ (X^ ∈^

A, Y^ ∈

B) =^ P

(X^ ∈^ A

)^ P^ (Y^ ∈

B)

In other words,

X^ and

Y^ are independent if, for all

A^ and^

B,

the events

E=^ A^

{X^ ∈^ A

}^ and^ F

=^ {Y^ b ∈^ B}^ are indepen-

dent. Note it is sufficient to restrict oneself to the set of thetype^ EA

=^ {X^

≤^ a}^ and

F=^ {b^

Y^ ≤^ b}

. Hence

independence between

X^ and

Y^ holds if and only if

P^ (X^ ≤

a, Y^ ≤

b) =^ P

(X^ ≤^ a

)^ P^ (Y^ ≤

b)

Random Variable and Expectation: Part III

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Independent Random Variables or equivalently

F^ (a, b) =

F(a)X^

F(b)Y^

When^

X^ and

Y^ are discrete random variables, the

condition of independence can be replaced by

p(x, y) =

p(x)^ p

(y)^

for all^

x, y

When^

X^ and

Y^ are continuous random variables, the

condition of independence can be replaced by

f^ (x, y) =

f(x)X^

f(y)^ Y^

for all^

x, y

Random Variable and Expectation: Part III

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Independent Random Variables

Hence the two lifetimes are independent. b). We have^ P^ (X <

3)^ =

∫^3 x e^0

−x^ dx^ = 1

− − 4 e

3 =^ P^ (

Y <^ 3)

The probability that the lifetime of at least onecomponent exceeds 3 is^1 −

P^ (X <

3 , Y <

3)^ =^

1 −^ P^ (

X <^ 3)

P^ (Y <

=^1 −

( 1 −^4

)^2 − 3 e

≈^.^3586365122

Random Variable and Expectation: Part III

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Conditional Distributions

Discrete Case Suppose

X^ and

Y^ be two discrete random variables

with joint probability mass function

p(x, y)

Then,^

the conditional distribution of

X^ given

Y^ =^ y

is

defined by

p(xX|Y^

|y) =^ P

(X^ =^ x

|Y^ =^ y

p(x, y) =

) , p(y)

where

p(y)^ >

Similarly, we define

the conditional distribution of

Y

given^ X

=^ x^ is p(yY^ |X^

|x) =^ P

(Y^ =^ y

|X^ =^ x

p(x, y) =

) , p(x)

where

p(x)^ >

Random Variable and Expectation: Part III 0.

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