Understanding Joint Probability Distributions in Statistics
Joint probability distributions are crucial in analyzing the simultaneous behavior of random variables. They can be described using mass functions for discrete variables and density functions for continuous variables. This concept is fundamental in probability and statistics, aiding in calculating probabilities and making informed decisions in various fields.
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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. 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Note: If X and Y are 2 discrete random variables, this distribution can be described with a joint probability mass function. If X and Y are continuous, this distribution can be described with a joint probability density function. 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Two Discrete Random Variables: If X and Y are discrete, with ranges ?? and ?? , respectively, the joint probability mass function is p(x, y) = P(X = x and Y = y), x ?? , y ??. 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
in the discrete case, 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Two Continuous Random Variables: If X and Y are continuous, the joint probability density function is a function f(x,y) that produces probabilities: ? ?,? ? = ? ?,? ???? A 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
in the continuous case, b d ( ) = 4) , ( , ) f x y dydx P a x b c y d a c 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Example: Suppose we have the following joint mass function Y -2 0 K 5 X 1 3 0.15 0.20 0.20 0.15 0.05 Find the value of k? 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Answer: Using ?(?,?) = 1 ? ? We get 0.15 + 0.20 + ? + 0.05 + 0.20 + 0.15 = 1 0.75 + ? = 1 ? = 1 0.75 = 0.25 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Example: Suppose we have the following joint density function 6 x y 0 2 , 2 4 x y = ( , ) f x y 8 0 . . OW 1) Prove that ?(?,?) is a joint probability function? 2) Calculate ? ? 2 3,? 5 2 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Answer: 1) ( , f x y ) 0 6 x y 4 2 = ( , ) f x y dxdy 8 2 0 1 8 4 2 ( ) = 6 x y dx dy 2 0 2 1 8 2 x 4 = 6 x yx dy 2 2 0 1 8 (2) 2 2 4 = 6(2) (2) 0 y dy 2 1 8 110 8 1 40 16 8 4 ( ) = 10 2 y dy 2 ( ) ( ) 4 = = 10(4) (4) 10(2) (2) 2 2 2 y y 2 1 8 ( ) ( ) ( ) 8 = = = 20 4 1 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
2 5 3 2 2 3 5 2 6 x y = 2) , P x y dydx 8 0 2 41 288 Prove that? = = 0.142 Another example see Ex 3.15 on page 96 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
The marginal distributions 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Example: Suppose we have the following joint mass function Y -2 0 5 X 1 3 0.15 0.20 0.25 0.05 0.20 0.15 Find the marginal distributions of X and Y? 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Answer: -2 0 5 Sum Y X 0.15 0.25 0.20 0.6 1 0.20 0.05 0.15 0.4 3 Sum 0.35 0.30 0.35 1 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
So x 1 3 Sum The marginal distribution of X 0.6 0.4 1 ( ) f x -2 0 5 Sum y ( ) f y 0.35 0.30 0.35 1 The marginal distribution of Y 15 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Example: Suppose we have the following joint density function ( ) = + ( , ) f x y , 0 1, 0 2 c x y x y Find the value of c ? Find the marginal distributions of X and Y? 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Answer: 2 1 ( ) = + = 1) ( , ) 1 1 f x y dxdy c x y dxdy 0 0 1 3 1 3 ( ) = = + ( , ) c f x y x y 1 3 2 ( ) = = + 2) ( ) ( , ) f x f x y x y dy 0 y 2 3 ( ) = + ( ) 1 f x x 1 3 1 ( ) = = + ( ) y ( , ) f f x y x y dx 0 x 1 3 1 2 = + ( ) y f y 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
conditional probability distribution 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Example: 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Solution: 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Another example see Ex 3.20 on page 100 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Statistical Independence 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Example: Suppose we have the following joint distribution 3 , 0 , 0 3 e e x OW y x y = ( , ) f x y 0 , . . Prove that X and Y are independent? 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
= ( , ) f x y ( ) ( ) f y f x = = = 1) ( ) ( , ) f x y dy 3 3 3 3 f x e e dy e e dy x y x y 0 0 3 e y = = 0 1 = 3 .....................(1) e e e x x x 3 0 = = = 2) ( ) f y ( , ) f x y dx 3 3 3 3 e e dx e e dx x y y x 0 0 = = 0 1 = 3 3 3 ... ............(2) 3 3 3 e e e e y x y y 0 (1) (2) From and = ( , ) f x y ( ) ( ) f y f x 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Notes: if X and Y are independent, then 1) ( , ) f x y = ( ) ( ) f y f x ( ( ) ) = 2) ( ) f x y f x = 3) ( ) f y f y x 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Example: Suppose we have the following joint distribution ( ) ( , ) 8 f x y k x y = , 0 4 ,1 3 x y Find: 1)The value of k ( ( ) ( ) 2) ( ) , f x ( ) f y 3) , f y x f x y ) 4) ( 3) 5) 3 2 P x P x y 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
Solution: 3 4 = = 1) ( , ) 1 (8 ) 1 f x y dxdy k x y dxdy 1 0 3 4 = (8 ) 1 k x y dx dy 1 0 4 2 x 3 = 8 1 k x xy 2 1 0 2 4 2 3 = 8(4) 4 1 k y dy 1 3 ( ) + = 4 24 1 k y dy 1 3 2 4 y + = 24 1 k y 2 1 1 32 1 32 ( ) = = = 32 1 ( , ) (8 ) k k f x y x y 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
= 2) ( ) x ( , x y dy ) f f 1 32 3 ( ) = ( ) x 8 f x y dy 1 1 32 ( ) = ( ) x 12 2 , 0 4 f x x = ( ) y ( , x y dx ) f f 1 32 4 ( ) = ( ) y 8 f x y dx 0 1 32 ( ) = ( ) y 24 4 , 1 3 f y y 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
1 32 ( ) 8 x y ( ) 8 x y ( , ) ( ) f x f x y ( ) = = = 3) f y x ( ) 1 32 12 2 x ( ) 12 2 x 1 32 1 32 ( ) 8 x y ( ) 8 x y ( , ) ( ) f y f x y ( ) = = = f x y ( ) 24 4 y ( ) 24 4 y 27 32 3 = = 4) ( 3) ( ) P x f x dx 0 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili
( 3, 2) P x y ( ) = 5) 3 2 P x y ( 2) p y 1 32 30 64 2 3 2 3 ( ) = = = ( 3, 2) ( , ) f x y dxdy 8 P x y x y dxdy 1 0 1 0 1 32 18 32 2 2 ( ) = = 24 4 = ( 2) ( ) f y dy p y y dy 1 1 5 6 ( ) = 3 2 P x y 503 STAT - Probability and Statistics for Engineers and Scientists Dr. Mansour Shrahili