Transcript of Lecture 28 Dr. MUMTAZ AHMED MTH 161: Introduction To Statistics.
- Slide 1
- Lecture 28 Dr. MUMTAZ AHMED MTH 161: Introduction To
Statistics
- Slide 2
- Review of Previous Lecture In last lecture we discussed:
Finding Area Under Normal Curve using MS-Excel Normal Approximation
to Binomial Distribution Central Limit Theorem Related examples
2
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- Objectives of Current Lecture In the current lecture: Joint
Distributions Moment Generating Functions Covariance Related
Examples 3
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- Joint Distributions 4
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- Joint Distributions Types of Joint Distribution: Discrete
Continuous Mixed A bivariate distribution may be discrete when The
possible values of (X,Y) are finite or countably infinite. It is
continuous if (X,Y) can assume all values in some non-countable set
of plane. It is said to be mixed if one r.v. is discrete and other
is continuous. 5
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- Discrete Joint Distributions 6
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- Bivariate Distributions Joint Probability Function also called
Bivariate Probability Function 7 X\Yy1y1 y2y2 ynyn P(X=x j ) x1x1
f(x 1,y 1 )f(x 1,y 2 )f(x 1,y n )g(x 1 ) x2x2 f(x 2,y 1 )f(x 2,y 2
)f(x 2,y n )g(x 2 ) xmxm f(x m,y 1 )f(x m,y 2 )f(x m,y n )g(x 3 )
P(Y=y j )h(y 1 )h(y 2 )h(y n )1
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- Bivariate Distributions Marginal Probability Functions:
Marginal Distribution of X Marginal Distribution of Y 8 X\Yy1y1
y2y2 ynyn P(X=x j ) x1x1 f(x 1,y 1 )f(x 1,y 2 )f(x 1,y n )g(x 1 )
x2x2 f(x 2,y 1 )f(x 2,y 2 )f(x 2,y n )g(x 2 ) xmxm f(x m,y 1 )f(x
m,y 2 )f(x m,y n )g(x 3 ) P(Y=y j )h(y 1 )h(y 2 )h(y n )1
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- Bivariate Distributions Conditional Probability Functions:
Conditional Probability of X/Y Conditional Probability of Y/X 9
X\Yy1y1 y2y2 ynyn P(X=x j ) x1x1 f(x 1,y 1 )f(x 1,y 2 )f(x 1,y n
)g(x 1 ) x2x2 f(x 2,y 1 )f(x 2,y 2 )f(x 2,y n )g(x 2 ) xmxm f(x m,y
1 )f(x m,y 2 )f(x m,y n )g(x 3 ) P(Y=y j )h(y 1 )h(y 2 )h(y n
)1
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- Bivariate Distributions Independence: Two r.v.s X and Y are
said to be independent iff for all possible pairs of values (x i, y
j ), the joint probability function f(x,y) can be expressed as the
product of the two marginal probability functions. 10
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- Bivariate Distributions Example: An urn contains 3 black, 2 red
and 3 green balls and 2 balls are selected at random from it. If X
is the number of black balls and Y is the number of red balls
selected, then find the joint probability distribution of X and Y.
Solution: Total Balls=3black+2red+3green=8 balls Possible values of
both X & Y are={0,1,2} The Joint Frequency Distribution 11
X\Y012g(x) 03/286/281/2810/28 19/286/28015/28 23/2800
H(y)15/2812/281/281
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- Bivariate Distributions The Joint Frequency Distribution P(X=0
|Y=1)=? 12 X\Y012g(x) 03/286/281/2810/28 19/286/28015/28 23/2800
H(y)15/2812/281/281 P(X+Y