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Lecture 28 Dr. MUMTAZ AHMED MTH 161: Introduction To Statistics

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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