Joint Probability DistributionsOutlines Two Discrete/Continuous Random Variables

Joint Probability Distributions Marginal Probability Distributions Conditional Probability Distributions Independence

Multiple Discrete/Continuous Random Variables Joint Probability Distributions Multinomial Probability Distribution

Covariance and Correlation Bivariate Normal Distribution Linear Combination of random variables

Joint Probability Distributions In general, if X and Y are two random variables,

the probability distribution that defines their simultaneous behavior is called a joint probability distribution.

For example: X : the length of one dimension of an injection-molded part, and Y : the length of another dimension. We might be interested in

P(2.95 X 3.05 and 7.60 Y 7.80).

Two Discrete Random Variables Joint Probability Distributions Marginal Probability Distributions Conditional Probability Distributions Independence

Joint Probability Distributions The joint probability distribution of two

random variables =bivariate probability distribution.

The joint probability distribution of two discrete random variables is usually written as P(X=x, Y=y).

Marginal Probability Distributions Marginal Probability Distribution: the

individual probability distribution of a random variable.

Marginal Probability Distributions Example: The marginal probability

distribution for X and Y. y=num

ber of times city name is stated

x=number of bars of signal strength

1 2 3 Marginal probability distribution of Y

4 0.15 0.1 0.05 0.3

3 0.02 0.1 0.05 0.17

2 0.02 0.03 0.2 0.25

1 0.01 0.02 0.25 0.28

0.2 0.25 0.55

Marginal probability distribution of X

P(X=3)

Conditional Probability Distributions When two random variables are defined

in a random experiment, knowledge of one can change the probabilities of the other.

Conditional Mean and Variance

Conditional Mean and Variance Example: From the previous example,

calculateP(Y=1|X=3), E(Y|1), and V(Y|1).

454.055.0/25.0)3(/)1,3(

)3(/)1,3()3|1(

,

xyx ff

XPYXPXYP

748.0

75.0)55.34(1.0)55.33(1.0)55.32(05.0)55.31(

)()()1|(

55.3)75.0(4)1.0(3)1.0(2)05.0(1

)()1|(

2222

1|2

|

1|

yYxY

yY

yfyYV

yyfYE

Independence

In some random experiments, knowledge of the values of X does not change any of the probabilities associated with the values for Y.

If two random variables are independent, then

Multiple Discrete Random Variables Joint Probability Distributions Multinomial Probability Distribution

Joint Probability Distributions In some cases, more than two random

variables are defined in a random experiment.

Marginal probability mass function

Joint Probability Distributions Mean and Variance

Joint Probability Distributions Conditional Probability Distributions

Independence

Multinomial Probability Distribution A joint probability distribution for multiple

discrete random variables that is quite useful in an extension of the binomial.

Multinomial Probability Distribution Example: Of the 20 bits received, what is the probability

that 14 are Excellent, 3 are Good, 2 are Fair, and 1 is Poor? Assume that the classifications of individual bits are independent events and that the probabilities of E, G, F, and P are 0.6, 0.3, 0.08, and 0.02, respectively.

One sequence of 20 bits that produces the specified

numbers of bits in each class can be represented as: EEEEEEEEEEEEEEGGGFFP

P(EEEEEEEEEEEEEEGGGFFP)= The number of sequences (Permutation of similar objects)=

912314 10708.202.008.03.06.0

2325600!1!2!3!14

!20

0063.010708.22325600)1,'2,'3,'14( 9 PsFsGsEP

Two Continuous Random Variables Joint Probability Distributions Marginal Probability Distributions Conditional Probability Distributions Independence

Joint Probability Distributions

Joint Probability Distributions Example: X: the time until a computer server connects to

your machine , Y: the time until the server authorizes you as a valid user. Each of these random variables measures the wait from a common starting time and X <Y. Assume that the joint probability density function for X and Y is

The probability that X<1000 and Y<2000 is:

yxyxyxf XY ),002.0001.0exp(106),( 6

Marginal Probability Distributions Similar to joint discrete random

variables, we can find the marginal probability distributions of X and Y from the joint probability distribution.

Marginal Probability Distributions Example: For the random variables in the previous

example, calculate the probability that Y exceeds 2000 milliseconds.

Conditional Probability Distributions

Conditional Probability Distributions Example: For the random variables in the previous example,

determine the conditional probability density function for Y given that X=x

Determine P(Y>2000|x=1500)

0)(,)(

),()(| xffor

xf

yxfyf X

X

XYxY

))(( | yf xY

Conditional Probability Distributions Mean and Variance

Conditional Probability Distributions Example: For the random variables in the previous

example, determine the conditional mean for Y given that x=1500

Independence

Independence

Example: Let the random variables X and Y denote the lengths of two dimensions of a machined part, respectively.

Assume that X and Y are independent random variables, and the distribution of X is normal with mean 10.5 mm and variance 0.0025 (mm)2 and that the distribution of Y is normal with mean 3.2 mm and variance 0.0036 (mm)2.

Determine the probability that 10.4 < X < 10.6 and 3.15 < Y < 3.25.

Because X,Y are independent

Multiple Continuous Random Variables

Multiple Continuous Random Variables Marginal Probability

Multiple Continuous Random Variables Mean and Variance

Independence

Covariance and Correlation

When two or more random variables are defined on a probability space, it is useful to describe how they vary together.

It is useful to measure the relationship between the variables.

Covariance

Covariance is a measure of linear relationship between the random variables.

\

The expected value of a function of two random variables

h(X, Y ).

Covariance

YXYXXY

YXYXYXXYXY

YXXYyYXXYX

YXY

XY

XYXXYX

XYYXYX

XYYXXY

XYEdxdyyxxyf

dxdyyxxyfXYEinSubstitute

dxdyyxfxanddxdyyxyfinSubstitute

dxdyyxyfyEyyhFor

dxdyyxfyhyhEFrom

dxdyyxyfdxdyyxyf

Now

dxdyyxfxyxy

dxdyyxfyxXYE

)(),(

),()])([(),1(

),(,),(),2(

),()(;)(

),()())((

)2(),(),(

)1(),(][

),())(()])([(

Covariance

Covariance

Example: For the discrete random variables X, Y with the joint distribution shown in Fig. Determine

XYXY and

Correlation

The correlation is a measure of the linear relationship between random variables.

Easier to interpret than the covariance.

Correlation

For independent random variables

Correlation

Example: Two random variables , calculate the covariance and correlation between X and Y.

xyyxf XY 16

1),(

Bivariate Normal Distribution

Correlation

Bivariate Normal Distribution Marginal distributions

Dependence

Bivariate Normal Distribution Conditional probability

)1( 222|

|

YxY

X

Y

X

YXYxY x

Bivariate Normal DistributionEx. Suppose that the X and Y dimensions of an injection-modeled

part have a bivariate normal distribution with

Find the P(2.95<X<3.05,7.60<Y<7.80)

8.0,70.7,00.3,08.0,04.0 yxyx

Bivariate Normal Distribution Ex. Let X, Y : milliliters of acid and base needed for

equivalence, respectively. Assume X and Y have a bivariate normal distribution with

Covariance between X and Y Marginal probability distribution of X P(X<116) P(X|Y=102) P(X<116|Y=102)

6.0,100,120,2,5 yxyx

Linear Combination of random variables

Linear Combination of random variables Mean and Variance

Linear Combination of random variablesEx. A semiconductor product consists of 3 layers. The

variances in thickness of the first, second, and third layers are 25,40,30 nm2 . What is the variance of the thickness of the final product?

Let X1, X2, X3, and X be random variables that denote the thickness of the respective layers, and the final product.

V(X)=V(X1)+V(X2)+V(X3)=25+40+30=95 nm2

Homework

1. The time between surface finish problems in a galvanizing process is exponentially distributed with a mean of 40 hours. A single plant operates three galvanizing lines that are assumed to operate independently.a) What is the probability that none of the lines experience a surface finish

problem in 40 hours of operation?

b) What is the probability that all three lines experience two surface finish problems between 20 and 40 hours after starting the operation?

2. Suppose X and Y have a bivariate normal distribution with

Determine the following:

a) P(2.95<X<3.05)

b) P(7.60<Y<7.80)

c) P(2.95<X<3.05,7.60<Y<7.80)

.0,70.7,00.3,08.0,04.0 yxyx