Joint Probability Distributions Outlines Two Discrete/Continuous Random Variables Joint...

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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 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=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=1X=3), E(Y1), and V(Y1).
454.055.0/25.0)3(/)1,3(
)3(/)1,3()31(
,
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
12

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>2000x=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 injectionmodeled
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(XY=102) P(X<116Y=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