Joint Probability Distributions
Outlines


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 injectionmolded 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=number
of times
1
city name
is stated
x=number of bars of signal strength
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
P(X=3)
Marginal probability distribution of X
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, calculate
P(Y=1|X=3), E(Y|1), and V(Y|1).

P(Y  1 | X  3)  P( X  3, Y  1) / P( X  3)
 f x , y (3,1) / f x (3)  0.25 / 0.55  0.454
E (Y | 1)   yfY |1 ( y )
y
 1(0.05)  2(0.1)  3(0.1)  4(0.75)  3.55
V (Y | 1)   ( y  Y | x ) 2 fY |1 ( y )
y
 (1  3.55) 2 0.05  (2  3.55) 2 0.1  (3  3.55) 2 0.1  (4  3.55) 2 0.75
 0.748
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)= 0.6140.330.0820.021  2.708 10 9
20!
 2325600
The number of sequences (Permutation of similar objects)=
14!3!2!1!
 P(14E ' s,3G' s,2F ' s,1P)  2325600  2.708 109  0.0063
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
f XY ( x, y)  6 106 exp( 0.001x  0.002 y), x  y

The probability that X<1000 and Y<2000 is:
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 ( f Y | x ( y ))
fY | x ( y ) 

f XY ( x, y )
,
f X ( x)
Determine P(Y>2000|x=1500)
for
f X ( x)  0
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
 
  (x  
E[(Y  Y )( X   X )] 
X
)( y  Y ) f XY ( x, y )dxdy
  
 

  [ xy  
X
y  xY   X Y ] f XY ( x, y )dxdy (1)
  
Now
 


yf
(
x
,
y
)
dxdy


yf
(
x
,
y
)
dxdy


X
XY
X
XY


  
 

 
( 2)
 
From E (h( y )) 
  h( y ) f
XY
( x, y )dxdy
  
 
For h( y )  y; E ( y ) 
  yf
XY
( x, y )dxdy  Y
  
 
Substitute in (2),
 
X
yf XY ( x, y )dxdy   X Y , and
  
 
  x
y
f XY ( x, y )dxdy   X Y
  
Substitute in (1), E[(Y  Y )( X   X )] 
 
  xyf
XY
( x, y )dxdy   X Y   X Y   X Y
  
 

  xyf
  
XY
( x, y )dxdy   X Y  E ( XY )  X Y
Covariance
Covariance

Example: For the discrete random variables X, Y with the joint distribution
shown in Fig. Determine  XY and  XY
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
and correlation between X and Y.
f XY ( x, y ) 
1
xy ,
16
calculate the covariance
Bivariate Normal Distribution
Correlation
Bivariate Normal Distribution

Marginal distributions

Dependence
Bivariate Normal Distribution

Conditional probability
Y | x  Y   X 
Y Y

x
X X
 Y2| x   Y2 (1   2 )
Bivariate Normal Distribution
Ex. Suppose that the X and Y dimensions of an injection-modeled part have a bivariate
normal distribution with  x  0.04,  y  0.08,  x  3.00,  y  7.70,   0.8
Find the P(2.95<X<3.05,7.60<Y<7.80)
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
 x  5,  y  2,  x  120,  y  100,   0.6

Covariance between X and Y

Marginal probability distribution of X

P(X<116)

P(X|Y=102)

P(X<116|Y=102)
Linear Combination of random
variables
Linear Combination of random
variables

Mean and Variance
Linear Combination of random
variables
Ex. 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.
2.
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?
Suppose X and Y have a bivariate normal distribution with
 x  0.04,  y  0.08,  x  3.00,  y  7.70,   0. 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)