Random Vector

Martina Litschmannovámartina.litschmannova@vsb.cz

K210

Random Vectors

An k- dimensional random vector is a function X = that associates a vector of real numbers with each element is a random variable.

For example: A semiconductor chip is divided into ‘M’ regions. For the random

experiment of finding the number of defects and their locations, let denote the number of defects in ith region. Then is a discrete random vector.

In a random experiment of selecting a student’s name, let = height of ith student in inches and = weight of ith student in pounds. Thenis a continuous random vector.

We're going to focus on 2-dimensional distributions (i.e. random vector consists only of two random variables) but higher dimensions (more than two variables) are also possible.

2 – dimensional Random Vectors

Joint Probability Distribution

Joint distribution for random vector defines the probability of events defined in terms of both X and Y.

Joint cumulative distribution function for is given by.

Joint cumulative distribution function

Properties of joint CDF:

1. ,2. ,3. ,4. is nondecreasing in each variable,5. is continous from the left in each variable.

Joint cumulative distribution function for is given by.

Discrete Joint Probability Distributions

The probability function, also known as the probability mass function for a joint probability distribution is defined such that:

.

Discrete Joint Probability Distributions

Probability Mass Function for a Joint Probability Distribution :

Properties of p.m.f.:1. only for a finite or countable set of values 2. ,

3. ,4. If X, Y are independent

X\Y y1 y2 ... yn2

x1 p(x1, y1) p(x1, y2) ... p(x1, yn2)x2 p(x2, y1) p(x2, y2) ... p(x2, yn2)

...xn1 p(xn1, y1) p(xn1, y2) ... p(xn1, yn2)

Table of joint probabilities

1. A random experiment consists of tossing coin (X) and flipping die (Y). Find probability mass function for a joint probability distribution of a random vector .

X/Y 1 2 3 4 5 60 (head) 1/12 1/12 1/12 1/12 1/12 1/121 (tail) 1/12 1/12 1/12 1/12 1/12 1/12

1. A random experiment consists of tossing coin (X) and flipping die (Y). Find probability mass function for a joint probability distribution of a random vector .

X/Y 1 2 3 4 5 60 (head) 1/12 1/12 1/12 1/12 1/12 1/121 (tail) 1/12 1/12 1/12 1/12 1/12 1/12

1. A random experiment consists of tossing coin (X) and flipping die (Y). Find probability mass function for a joint probability distribution of a random vector .

X/Y 1 2 3 4 5 60 (head) 1/12 1/12 1/12 1/12 1/12 1/121 (tail) 1/12 1/12 1/12 1/12 1/12 1/12

1

control cell

2. Probability mass function for a joint probability distribution of a random vector is given as:

X/Y -2 0 1 2 -1 0,13 0,11 0,07 0 0 0 0,11 0 0,33 1 0,23 0,01 0 0,01 1

Find:

Continous Joint Probability Distributions

,

where is Joint Probability Density Function.

Properties of Joint Probability Density Function:1. ,2. ,3. If exists, pak ,4. .

3. Find the constant c so that function can be a joint probability density function of a random vector .

That the function can be a joint probability density function of a random vector , i condition

.

Marginal probability distributions

Obtained by summing or integrating the joint probability distribution over the values of the other random variable.

Discrete Random Vector,.

Continous Random Vector, ,.

4. A random experiment consists of tossing coin (X) and flipping die (Y). Probability mass function for a joint probability distribution of a random vector

X/Y 1 2 3 4 5 60 (head) 1/12 1/12 1/12 1/12 1/12 1/121 (tail) 1/12 1/12 1/12 1/12 1/12 1/12

1

Find Marginal Probability Mass Functions and

4. A random experiment consists of tossing coin (X) and flipping die (Y). Probability mass function for a joint probability distribution of a random vector

X/Y 1 2 3 4 5 6 0 (head) 1/12 1/12 1/12 1/12 1/12 1/12 6/121 (tail) 1/12 1/12 1/12 1/12 1/12 1/12 6/12

2/12 2/12 2/12 2/12 2/12 2/12 1

Find Marginal Probability Mass Functions and

4. A random experiment consists of tossing coin (X) and flipping die (Y). Probability mass function for a joint probability distribution of a random vector

X/Y 1 2 3 4 5 6 0 (head) 1/12 1/12 1/12 1/12 1/12 1/12 6/121 (tail) 1/12 1/12 1/12 1/12 1/12 1/12 6/12

2/12 2/12 2/12 2/12 2/12 2/12 1

Marginal Probability Mass Functions and

Y 1 2 3 4 5 62/12 2/12 2/12 2/12 2/12 2/12

X 0 (head) 1 (tail) 6/12 6/12

4. A random experiment consists of tossing coin (X) and flipping die (Y). Probability mass function for a joint probability distribution of a random vector

X/Y 1 2 3 4 5 6 0 (head) 1/12 1/12 1/12 1/12 1/12 1/12 1/21 (tail) 1/12 1/12 1/12 1/12 1/12 1/12 1/2

1/6 1/6 1/6 1/6 1/6 1/6 1

Y 1 2 3 4 5 61/6 1/6 1/6 1/6 1/6 1/6

X 0 (head) 1 (tail) 1/2 1/2

Marginal Probability Mass Functions and

5. A certain farm produces two kinds of eggs on any given day; organic and non-organic. Let these two kinds of eggs be represented by the random variables X and Y respectively. Given that the joint probability density function of these variables is given by

Find: a) marginal density functions and

5. A certain farm produces two kinds of eggs on any given day; organic and non-organic. Let these two kinds of eggs be represented by the random variables X and Y respectively. Given that the joint probability density function of these variables is given by

Find: b)

5. A certain farm produces two kinds of eggs on any given day; organic and non-organic. Let these two kinds of eggs be represented by the random variables X and Y respectively. Given that the joint probability density function of these variables is given by

Find: c)

Conditional probability distributions

Conditional Probability Distributions arise from joint probability distributions where by we need to know that probability of one event given that the other event has happened, and the random variables behind these events are joint.

Discrete Random Vector0,.

Continous Random Vector0,.

6. Joint and marginal probability distribution of a random vector is given as:

X/Y -2 0 1 2 -1 0,13 0,11 0,07 0 0,31 0 0 0,11 0 0,33 0,441 0,23 0,01 0 0,01 0,25 0,36 0,23 0,07 0,34 1

Find:

7. Joint probability density function of is given by

Find: a) conditional density function

7. Joint probability density function of is given by

Find: b) conditional density function

Conditional expected value (expectation)

Conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution.

Discrete random vector:

Continous random vector:

Conditional variance

Conditional variance is the variance of a conditional probability distribution.

8. Joint and marginal probability distribution of a random vector is given as:

X/Y -2 0 1 2 -1 0,13 0,11 0,07 0 0,31 0 0 0,11 0 0,33 0,441 0,23 0,01 0 0,01 0,25 0,36 0,23 0,07 0,34 1

Find:

9. Joint probability density function of is given by

Find: a) E b) D.

Independence

Two random variables X and Y are independent if

Discrete Random Variables

Continous Random Vector

.

Measures of Linear Independence

Covariance:

Correlation coefficient:

is a scaled version of covariance

Covariance

Covariance:

Covariance matrix:

Correlation

Correlation:

… are positively correlated … are negatively correlated … are uncorrelated

Correlation matrix:

=1,000 = -1,000 =0,000 =0,934

=0,967 =0,857 =-0,143 =0,608

Correlation

10. Joint and marginal probability distribution of a random vector is given as:

X/Y -2 0 1 2 -1 0,13 0,11 0,07 0 0,31 0 0 0,11 0 0,33 0,441 0,23 0,01 0 0,01 0,25 0,36 0,23 0,07 0,34 1

Find:a) , b) , c) , d) Are random variable X, Y independent?e) Are random variable X, Y linear independent?

11. Joint probability density function of is given by

Marginal density functions are:

Find:a) , b) , c) , d) Are random variable X, Y independent?e) Are random variable X, Y linear independent?

Study materials :

http://homel.vsb.cz/~bri10/Teaching/Bris%20Prob%20&%20Stat.pdf (p. 64 - p.70)