Jointly Distributed Random Variables

Consider tossing a fair die twice. Then the outcomes would be

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)· · · · · · · · · · · · · · · · · ·

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

and the probability for each outcome is 136 .

If we define two random variables by X = the outcome of the

first toss and Y = the outcome of the second toss, thenthe outcome for this experiment (two tosses) can be describe bythe random pair (X , Y ), and the probability for any possible valueof that random pair (x , y) is 1

36 .

Jointly Distributed Random Variables

Consider tossing a fair die twice. Then the outcomes would be

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)· · · · · · · · · · · · · · · · · ·

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

and the probability for each outcome is 136 .

If we define two random variables by X = the outcome of the

first toss and Y = the outcome of the second toss, thenthe outcome for this experiment (two tosses) can be describe bythe random pair (X , Y ), and the probability for any possible valueof that random pair (x , y) is 1

36 .

Jointly Distributed Random Variables

Consider tossing a fair die twice. Then the outcomes would be

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)· · · · · · · · · · · · · · · · · ·

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

and the probability for each outcome is 136 .

If we define two random variables by X = the outcome of the

first toss and Y = the outcome of the second toss,

thenthe outcome for this experiment (two tosses) can be describe bythe random pair (X , Y ), and the probability for any possible valueof that random pair (x , y) is 1

36 .

Jointly Distributed Random Variables

Consider tossing a fair die twice. Then the outcomes would be

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)· · · · · · · · · · · · · · · · · ·

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

and the probability for each outcome is 136 .

If we define two random variables by X = the outcome of the

first toss and Y = the outcome of the second toss, thenthe outcome for this experiment (two tosses) can be describe bythe random pair (X , Y ),

and the probability for any possible valueof that random pair (x , y) is 1

36 .

Jointly Distributed Random Variables

Consider tossing a fair die twice. Then the outcomes would be

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)· · · · · · · · · · · · · · · · · ·

(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)

and the probability for each outcome is 136 .

If we define two random variables by X = the outcome of the

first toss and Y = the outcome of the second toss, thenthe outcome for this experiment (two tosses) can be describe bythe random pair (X , Y ), and the probability for any possible valueof that random pair (x , y) is 1

36 .

Jointly Distributed Random Variables

DefinitionLet X and Y be two discrete random variables defined on thesample space S of an experiment. The joint probability massfunction p(x , y) is defined for each pair of numbers (x , y) by

p(x , y) = P(X = x and Y = y)

(It must be the case that p(x , y) ≥ 0 and∑

x

∑y p(x , y) = 1.)

For any event A consisting of pairs of (x , y), the probabilityP[(X , Y ) ∈ A] is obtained by summing the joint pmf over pairs inA:

P[(X , Y ) ∈ A] =∑ ∑(x ,y)∈A

p(x , y)

Jointly Distributed Random Variables

DefinitionLet X and Y be two discrete random variables defined on thesample space S of an experiment. The joint probability massfunction p(x , y) is defined for each pair of numbers (x , y) by

p(x , y) = P(X = x and Y = y)

(It must be the case that p(x , y) ≥ 0 and∑

x

∑y p(x , y) = 1.)

For any event A consisting of pairs of (x , y), the probabilityP[(X , Y ) ∈ A] is obtained by summing the joint pmf over pairs inA:

P[(X , Y ) ∈ A] =∑ ∑(x ,y)∈A

p(x , y)

Jointly Distributed Random Variables

Example (Problem 75)A restaurant serves three fixed-price dinners costing $12, $15, and$20. For a randomly selected couple dinning at this restaurant, letX = the cost of the man’s dinner and Y = the cost of

the woman’s dinner. If the joint pmf of X and Y is assumed tobe

yp(x , y) 12 15 20

12 .05 .05 .10x 15 .05 .10 .35

20 0 .20 .10

a. What is the probability for them to both have the $12 dinner?

b. What is the probability that they have the same price dinner?

c. What is the probability that the man’s dinner cost $12?

Jointly Distributed Random Variables

Example (Problem 75)A restaurant serves three fixed-price dinners costing $12, $15, and$20. For a randomly selected couple dinning at this restaurant, letX = the cost of the man’s dinner and Y = the cost of

the woman’s dinner. If the joint pmf of X and Y is assumed tobe

yp(x , y) 12 15 20

12 .05 .05 .10x 15 .05 .10 .35

20 0 .20 .10

a. What is the probability for them to both have the $12 dinner?

b. What is the probability that they have the same price dinner?

c. What is the probability that the man’s dinner cost $12?

Jointly Distributed Random Variables

Example (Problem 75)A restaurant serves three fixed-price dinners costing $12, $15, and$20. For a randomly selected couple dinning at this restaurant, letX = the cost of the man’s dinner and Y = the cost of

the woman’s dinner. If the joint pmf of X and Y is assumed tobe

yp(x , y) 12 15 20

12 .05 .05 .10x 15 .05 .10 .35

20 0 .20 .10

a. What is the probability for them to both have the $12 dinner?

b. What is the probability that they have the same price dinner?

c. What is the probability that the man’s dinner cost $12?

Jointly Distributed Random Variables

Example (Problem 75)A restaurant serves three fixed-price dinners costing $12, $15, and$20. For a randomly selected couple dinning at this restaurant, letX = the cost of the man’s dinner and Y = the cost of

the woman’s dinner. If the joint pmf of X and Y is assumed tobe

yp(x , y) 12 15 20

12 .05 .05 .10x 15 .05 .10 .35

20 0 .20 .10

a. What is the probability for them to both have the $12 dinner?

b. What is the probability that they have the same price dinner?

c. What is the probability that the man’s dinner cost $12?

Jointly Distributed Random Variables

Example (Problem 75)A restaurant serves three fixed-price dinners costing $12, $15, and$20. For a randomly selected couple dinning at this restaurant, letX = the cost of the man’s dinner and Y = the cost of

the woman’s dinner. If the joint pmf of X and Y is assumed tobe

yp(x , y) 12 15 20

12 .05 .05 .10x 15 .05 .10 .35

20 0 .20 .10

a. What is the probability for them to both have the $12 dinner?

b. What is the probability that they have the same price dinner?

c. What is the probability that the man’s dinner cost $12?

Jointly Distributed Random Variables

DefinitionLet X and Y be two discrete random variables defined on thesample space S of an experiment with joint probability massfunction p(x , y). Then the pmf’s of each one of the variables aloneare called the marginal probability mass functions, denoted bypX (x) and pY (y), respectively. Furthermore,

pX

(x) =∑y

p(x , y) and pY

(y) =∑x

p(x , y)

Jointly Distributed Random Variables

DefinitionLet X and Y be two discrete random variables defined on thesample space S of an experiment with joint probability massfunction p(x , y). Then the pmf’s of each one of the variables aloneare called the marginal probability mass functions, denoted bypX (x) and pY (y), respectively. Furthermore,

pX

(x) =∑y

p(x , y) and pY

(y) =∑x

p(x , y)

Jointly Distributed Random Variables

Example (Problem 75) continued:The marginal probability mass functions for the previous example iscalculated as following:

y

p(x , y) 12 15 20 p(x , ·)12 .05 .05 .10 .20

x 15 .05 .10 .35 .5020 0 .20 .10 .30

p(·, y) .10 .35 .55

Jointly Distributed Random Variables

Example (Problem 75) continued:The marginal probability mass functions for the previous example iscalculated as following:

y

p(x , y) 12 15 20 p(x , ·)12 .05 .05 .10 .20

x 15 .05 .10 .35 .5020 0 .20 .10 .30

p(·, y) .10 .35 .55

Jointly Distributed Random Variables

DefinitionLet X and Y be continuous random variables. A joint probabilitydensity function f (x , y) for these two variables is a functionsatisfying f (x , y) ≥ 0 and

∫∞−∞

∫∞−∞ f (x , y)dxdy = 1.

For any two-dimensional set A

P[(X , Y ) ∈ A] =

∫∫A

f (x , y)dxdy

In particular, if A is the two-dimensilnal rectangle{(x , y) : a ≤ x ≤ b, c ≤ y ≤ d}, then

P[(X , Y ) ∈ A] = P(a ≤ X ≤ b, c ≤ Y ≤ d) =

∫ b

a

∫ d

cf (x , y)dydx

Jointly Distributed Random Variables

DefinitionLet X and Y be continuous random variables. A joint probabilitydensity function f (x , y) for these two variables is a functionsatisfying f (x , y) ≥ 0 and

∫∞−∞

∫∞−∞ f (x , y)dxdy = 1.

For any two-dimensional set A

P[(X , Y ) ∈ A] =

∫∫A

f (x , y)dxdy

In particular, if A is the two-dimensilnal rectangle{(x , y) : a ≤ x ≤ b, c ≤ y ≤ d}, then

P[(X , Y ) ∈ A] = P(a ≤ X ≤ b, c ≤ Y ≤ d) =

∫ b

a

∫ d

cf (x , y)dydx

Jointly Distributed Random Variables

DefinitionLet X and Y be continuous random variables with joint pdff (x , y). Then the marginal probability density functions of Xand Y , denoted by f

X(x) and f

Y(y), respectively, are given by

fX

(x) =

∫ ∞−∞

f (x , y)dy for −∞ < x <∞

fY

(y) =

∫ ∞−∞

f (x , y)dx for −∞ < y <∞

Jointly Distributed Random Variables

DefinitionLet X and Y be continuous random variables with joint pdff (x , y). Then the marginal probability density functions of Xand Y , denoted by f

X(x) and f

Y(y), respectively, are given by

fX

(x) =

∫ ∞−∞

f (x , y)dy for −∞ < x <∞

fY

(y) =

∫ ∞−∞

f (x , y)dx for −∞ < y <∞

Jointly Distributed Random Variables

Example (variant of Problem 12)Two components of a minicomputer have the following joint pdffor their useful lifetimes X and Y :

f (x , y) =

{xe−(x+y) x ≥ 0 and y ≥ 0

0 otherwise

a. What is the probability that the lifetimes of both componentsexcceed 3?

b. What are the marginal pdf’s of X and Y ?

c. What is the probability that the lifetime X of the firstcomponent excceeds 3?

d. What is the probability that the lifetime of at least onecomponent excceeds 3?

Jointly Distributed Random Variables

Example (variant of Problem 12)Two components of a minicomputer have the following joint pdffor their useful lifetimes X and Y :

f (x , y) =

{xe−(x+y) x ≥ 0 and y ≥ 0

0 otherwise

a. What is the probability that the lifetimes of both componentsexcceed 3?

b. What are the marginal pdf’s of X and Y ?

c. What is the probability that the lifetime X of the firstcomponent excceeds 3?

d. What is the probability that the lifetime of at least onecomponent excceeds 3?

Jointly Distributed Random Variables

Example (variant of Problem 12)Two components of a minicomputer have the following joint pdffor their useful lifetimes X and Y :

f (x , y) =

{xe−(x+y) x ≥ 0 and y ≥ 0

0 otherwise

a. What is the probability that the lifetimes of both componentsexcceed 3?

b. What are the marginal pdf’s of X and Y ?

c. What is the probability that the lifetime X of the firstcomponent excceeds 3?

d. What is the probability that the lifetime of at least onecomponent excceeds 3?

Jointly Distributed Random Variables

Example (variant of Problem 12)Two components of a minicomputer have the following joint pdffor their useful lifetimes X and Y :

f (x , y) =

{xe−(x+y) x ≥ 0 and y ≥ 0

0 otherwise

a. What is the probability that the lifetimes of both componentsexcceed 3?

b. What are the marginal pdf’s of X and Y ?

c. What is the probability that the lifetime X of the firstcomponent excceeds 3?

d. What is the probability that the lifetime of at least onecomponent excceeds 3?

Jointly Distributed Random Variables

Example (variant of Problem 12)Two components of a minicomputer have the following joint pdffor their useful lifetimes X and Y :

f (x , y) =

{xe−(x+y) x ≥ 0 and y ≥ 0

0 otherwise

a. What is the probability that the lifetimes of both componentsexcceed 3?

b. What are the marginal pdf’s of X and Y ?

c. What is the probability that the lifetime X of the firstcomponent excceeds 3?

d. What is the probability that the lifetime of at least onecomponent excceeds 3?

Jointly Distributed Random Variables

Example (variant of Problem 12)Two components of a minicomputer have the following joint pdffor their useful lifetimes X and Y :

f (x , y) =

{xe−(x+y) x ≥ 0 and y ≥ 0

0 otherwise

a. What is the probability that the lifetimes of both componentsexcceed 3?

b. What are the marginal pdf’s of X and Y ?

c. What is the probability that the lifetime X of the firstcomponent excceeds 3?

d. What is the probability that the lifetime of at least onecomponent excceeds 3?