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Atatürk University
Introduction to Statistics
Prof. Dr. İrfan KAYMAZ
STATISTICS and PROBABILITY
Atatürk UniversityEngineering Faculty
Department of Mechanical Engineering
LECTURE: JOINT PROBABILITY DISTRIBUTIONS
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint Probability
Use joint probability mass functions and joint probability density functions to calculate probabilities.
Calculate marginal and conditional probability distributions from joint probability distributions.
Interpret and calculate covariances and correlations between random variables.
objectives of this lecture
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint Probability
Some random variables are not independent of each other, i.e., they tend to be related. Urban atmospheric ozone and airborne particulate matter
tend to vary together. Urban vehicle speeds and fuel consumption rates tend to
vary inversely.
The length (X) of a injection-molded part might not be independent of the width (Y). Individual parts will vary due to random variation in materials and pressure.
A joint probability distribution will describe the behavior of several random variables, say, X and Y. The graph of the distribution is 3-dimensional: x, y, and f(x,y).
objectives of this lecture
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: Signal BarsYou use your cell phone to check your airline reservation. The airline system
requires that you speak the name of your departure city to the voice recognition system.
• Let Y denote the number of times that you have to state your departure city.
• Let X denote the number of bars of signal strength on you cell phone.
Figure : Joint probability distribution of X and Y. The table cells are the probabilities. Observe that more bars relate to less repeating.
Once
Twice3 Times
4 Times
0.00
0.05
0.10
0.15
0.20
0.25
12
3
Pro
bab
ility
Cell Phone Bars
Bar Chart of Number of Repeats vs. Cell
Phone Bars1 2 3
1 0.01 0.02 0.25
2 0.02 0.03 0.20
3 0.02 0.10 0.05
4 0.15 0.10 0.05
x = number of bars
of signal strength
y = number of
times city
name is stated
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityJoint Probability Density Function Defined
Figure: Joint probability density function for the random variables Xand Y. Probability that (X, Y) is in the region R is determined by the volume of fXY(x,y) over the region R.
The joint probability density function for the continuous random variables X and Y, denotes as fXY(x,y), satisfies the following properties:
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityJoint Probability Mass Function Graph
Figure :Joint probability density function for the continuous random variables X and Y of different dimensions of an injection-molded part. Note the asymmetric, narrow ridge shape of the PDF – indicating that small values in the X dimension are more likely to occur when small values in the Y dimension occur.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: Server Access Time-1
Let the random variable X denote the time (msec’s) until a computerserver connects to your machine. Let Y denote the time until theserver authorizes you as a valid user. X and Y measure the wait from acommon starting point (x < y).
Figure: The joint probability density function of X and Y is nonzero over the shaded region where x < y.
0.001 0.002 6, for 0 and 6 10x y
XYf x y ke x y k
Compute the probability that X<1000 and Y<2000
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: Server Access Time-2
We verify that it integrates to 1 as follows:
0.001 0.002 0.002 0.001
0 0
0.0020.001 0.003
0 0
,
0.0030.002
10.003 1
0.003
x y y x
XY
x x
xx x
f x y dxdy ke dy dx k e dy e dx
ek e dx e dx
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: Server Access Time-3
Now calculate a probability:
Figure:Region of integration for the probability that X < 1000 and Y < 2000 is darkly shaded.
1000 2000
1000 2000
0.002 0.001
0
1000 0.002 40.001
0
1000
0.003 4 0.001
0
3 14
1000, 2000 ,
0.002
0.003
1 10.003
0.003 0.001
XY
x
y x
x
xx
x x
P X Y f x y dxdy
k e dy e dx
e ek e dx
e e e dx
e ee
0.003 316.738 11.578 0.915
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityMarginal Probability Distributions (continuous)
The marginal PDFs are used to makeprobability statements about one variable.
If the joint probability density function ofrandom variables X and Y is fXY(x,y), themarginal probability density functions of Xand Y are:
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: a simple function
Find the marginal distribution function for X and Y
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: a simple function
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: Server Access Time-1
For the random variables times in the previousexample, find the probability that Y exceeds 2000.
Integrate the joint PDF directly using the picture todetermine the limits.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: Server Access Time-2
2000
0 2000 2000
2000 , ,
Dark region left dark region right dark region
XY XY
x
P Y f x y dy dx f x y dy dx
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: Server Access Time-3
Alternatively, find the marginal PDF and then integrate that to find the desired probability.
0.001 0.002
0
0.002 0.001
0
0.0010.002
0
0.0010.002
3 0.002 0.001
0.001
1
0.001
6 10 1 for 0
y
x y
Y
y
y x
yx
y
yy
y y
f y ke
ke e dx
eke
eke
e e y
2000
3 0.002 0.001
2000
0.002 0.0033
2000 2000
4 63
2000
6 10 1
6 100.002 0.003
6 10 0.050.002 0.003
Y
y y
y y
P Y f y dy
e e dy
e e
e e
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityConditional Probability Density Function Defined
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: a simple function
Find the conditional distribution function X given Y
Find the probability that X<1 given that y=1/2
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityAnswer to a simple function
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: a simple function
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: Conditional Probability-1
From Example 5-2, determine the conditional PDF for Y given X=x.
0.001 0.002
0.0020.001
0.0020.001
0.003
0.001 0.002
0.003
0.002 0.002
0.002
0.002
0.003 for 0
,
0.003
0.002 for 0
x y
X
x
yx
x
x
x
x yXY
Y x x
X
x y
f x k e dy
eke
eke
e x
f x y kef y
ef x
e x y
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample 5-6: Conditional Probability-2
Now find the probability that Y exceeds 2000 given that X=1500:
1500
2000
0.002 1500 0.002
2000
0.0023
2000
43 1
2000 1500
0.002
0.0020.002
0.002 0.3680.002
Y
y
y
P Y X
f y dy
e
ee
ee e
Figure : The conditional PDF is nonzero on the solid line in the shaded region.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityMean & Variance of Conditional Random Variables
The conditional mean of Y given X = x, denoted as E(Y|x) or μY|x is:
The conditional variance of Y given X = x, denoted as V(Y|x) or σ2
Y|x is:
2
2 2
Y x Y x Y x Y x
y y
V Y x y f y dy y f y
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample: Conditional Mean & Variance
From Example 5-2 & 6, what is the conditional mean for Y given that x = 1500? Integrate by parts.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityCovariance
Covariance is a measure of the relationship between two random variables.
First, we need to describe the expected value of a function of two random variables. Let h(X, Y) denote the function of interest.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityCovariance Defined
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityCovariance Defined
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityExample
Find the covariance of X and Y
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityAnswer to Example
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityCovariance and Scatter Patterns
Figure: Joint probability distributions and the sign of cov(X, Y). Note that covariance is a measure of linear relationship. Variables with non-zero covariance are correlated.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityCorrelation (ρ = rho)
The between random variables X and Y,
denoted as , is
cov , (5-15)
Since 0 and 0,
and cov , have the same sign.
We say that
correlation
XY
XYXY
X Y
X Y
XY
XY
X Y
V X V Y
X Y
is normalized, so 1 1 (5-16)
Note that is dimensionless.
Variables with non-zero correlation are correlated.
XY
XY
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityIndependence Implies ρ = 0
If X and Y are independent random variables,
σXY = ρXY = 0
ρXY = 0 is necessary, but not a sufficient condition for independence.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk UniversityAtatürk University
Joint ProbabilityNext Lecture
Point Estimation….