Joint and Marginal Probability Distributions and their Functions, Study notes of Advanced Calculus

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Chapter 2 Multivariate Distributions

2.1 Distributions of Two Random Variables

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Bivariate random vector

Definition A random variable is a function from a sample space C to R.

Definition An n-dim random vector is a function from C to Rn. I (^) A 2 -dim random vector is also called a bivariate random variable. Remark: X = (X 1 , X 2 )′^ assigns to each element c of the sample space C exactly one ordered pair of numbers X 1 (c) = x 1 and X 2 (c) = x 2. Example 1 Height and weight of respondent. 2 Fuel consumption and hours on an engine.

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Joint probability mass function

Definition A joint probability mass function pX 1 ,X 2 (x 1 , x 2 ) = p(X 1 = x 1 , X 2 = x 2 ) (or p(x 1 , x 2 )) with space (x 1 , x 2 ) ∈ S has the properties that (a) 0 ≤ p(x 1 , x 2 ) ≤ 1 , (b)

(x 1 ,x 2 )∈S p(x^1 , x^2 ) = 1, (c) P [(X 1 , X 2 ) ∈ A] =

(x 1 ,x 2 )∈A p(x^1 , x^2 ).

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Example

A restaurant serves three fixed-price dinners costing $7, $9, and $10. For a randomly selected couple dinning at this restaurant, let X 1 = the cost of the man’s dinner and X 2 = the cost of the woman’s dinner. The joint pmf of X 1 and X 2 is given in the following table:

x 1 7 9 10 7 0. 05 0. 05 0. 10 x 2 9 0. 05 0. 10 0. 35 10 0. 00 0. 20 0. 10

I (^) What is the probability of P (X 1 ≥ 9 , X 2 ≤ 9)? 0. 60. I (^) Does man’s dinner cost more?

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Example

Let X 1 =Smaller die face, X 2 =Larger die face, when rolling a pair of two dice. The following table shows a partition of the sample space into 21 events. x 1 1 2 3 4 5 6 1 1 / 36 0 0 0 0 0 2 2 / 36 1 / 36 0 0 0 0 x 2 3 2 / 36 2 / 36 1 / 36 0 0 0 4 2 / 36 2 / 36 2 / 36 1 / 36 0 0 5 2 / 36 2 / 36 2 / 36 2 / 36 1 / 36 0 6 2 / 36 2 / 36 2 / 36 2 / 36 2 / 36 1 / 36

Find the marginal pmf’s.

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Expectation – discrete random variables

Definition Let Y = u(X 1 , X 2 ). Then, Y is a random variable and

E[u(X 1 , X 2 )] =

x 1

x 2

u(x 1 , x 2 )p(x 1 , x 2 )

under the condition that ∑

x 1

x 2

|u(x 1 , x 2 )|p(x 1 , x 2 )| < ∞

Example Find E(max{X 1 , X 2 }) for the restaurant problem. 9. 65.

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Joint density function

A joint density function fX 1 ,X 2 (x 1 , x 2 ) (or f (x 1 , x 2 )) with space (x 1 , x 2 ) ∈ S has the properties that (a) f (x 1 , x 2 ) > 0 , (b)

(x 1 ,x 2 )∈S f^ (x^1 , x^2 )dx^1 dx^2 = 1, (c) P [(X 1 , X 2 ) ∈ A] =

(x 1 ,x 2 )∈A f^ (x^1 , x^2 )dx^1 dx^2.

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Example

Let X 1 and X 2 be continuous random variables with joint density function

f (x 1 , x 2 ) =

4 x 1 x 2 for 0 < x 1 , x 2 < 1 0 otherwise.

1 Find P (1/ 4 < X 1 < 3 /4; 1/ 2 < X 2 < 1). 2 Find P (X 1 < X 2 ). 3 Find P (X 1 + X 2 < 1). Solution : ∫ (^1)

1 / 2

1 / 4

4 x 1 x 2 dx 1 dx 2 = 3/8 = 0. 375. ∫ (^1)

0

∫ (^) x 2

0

4 x 1 x 2 dx 1 dx 2 = 1/2 = 0. 5. ∫ (^1)

0

∫ (^1) −x 2

0

4 x 1 x 2 dx 1 dx 2 = 1/6 = 0. 167.

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Example

Let X 1 and X 2 be continuous random variables with joint density function

f (x 1 , x 2 ) =

cx 1 x 2 for 0 < x 1 < x 2 < 1 0 otherwise.

1 Find c. 2 Find P (X 1 + X 2 < 1). 3 Find marginal probability density function of X 1 and X 2.

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Solution:

We have c = 8 because ∫ (^1)

0

x 1

x 1 x 2 dx 1 dx 2 = 1/8 = 0. 125. ∫ (^1) / 2

0

∫ (^1) −x 1

x 1

8 x 1 x 2 dx 1 dx 2 = 1/6 = 0. 167.

For the marginal pdf, we have

fX 1 (x 1 ) =

x 1

8 x 1 x 2 dx 2 = 4x 1 − 4 x^31 ,

fX 2 (x 2 ) =

∫ (^) x 2

0

8 x 1 x 2 dx 1 = 4x^32.

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Expectation – continuous random variables

Let Y = u(X 1 , X 2 ). Then, Y is a random variable and

E[u(X 1 , X 2 )] =

x 1

x 2

u(x 1 , x 2 )f (x 1 , x 2 )dx 2 dx 1

under the condition that ∫

x 1

x 2

|u(x 1 , x 2 )|f (x 1 , x 2 )dx 2 dx 1 < ∞

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Example

Let X 1 and X 2 be continuous random variables with joint density function

f (x 1 , x 2 ) =

(36/5)x 1 x 2 (1 − x 1 x 2 ) for 0 < x 1 , x 2 < 1

0 otherwise.

Find E(X 1 X 2 ). Solution : ∫ (^1)

0

0

(x^21 x^22 (1 − x 1 x 2 ))dx 1 dx 2 = 0. 35.

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Example 2.1.5 & 2.1.

Let (X 1 , X 2 ) be a random vector with pdf

f (x 1 , x 2 ) =

8 x 1 x 2 0 < x 1 < x 2 < 1 0 elsewhere.

Let Y 1 = 7X 1 X^22 + 5X 2 and Y 2 = X 1 /X 2. Determine E(Y 1 ) and E(Y 2 ).

Discrete & Continuous R.V.

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