Post on 23-Jan-2016
description
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)