Understanding Joint Probability Distribution in Statistics

JITENDRA KUMAR PATI Associate Professor Dept. Of Math RECAP D I S C R E T E P R O B A B I L I T Y D I S C R E T E P R O B A B I L I T Y D I S T R I B U T I O N D I S T R I B U T I O N C O N T I N U O U S P R O B A B I L I T Y C O N T I N U O U S P R O B A B I L I T Y D I S T R I B U T I O N D I S T R I B U T I O N R E L A T E D P R O B L E M S R E L A T E D P R O B L E M S

L E C T U R E - 8 OUTLINE Joint Probability Distribution Discrete and continuous Distribution Relevant Properties Examples

JOINT PROBABILITY DISTRIBUTION Distributions of two or more random variables are of interest for two reasons: They occur in experiments in which we observe several random variables, for example, carbon content X and hardness Y of steel, amount of fertilizer X and yield of corn Y, height weight and blood pressure of persons, and so on. In this section we consider two random variables X and Y or, as we also say, a two dimensional random variable (X,Y). For(X,Y) the outcome of a trial is a pair of numbers X=x, Y=y briefly(X,Y)=(x,y) which we can plot as a point in the XY-plane. A discrete bivariate random variable (X, Y ) is an ordered pair of discrete random variables.

Let (X, Y ) be a bivariate random variable and let RX and RY be the range spaces of X and Y , respectively. A real-valued function Let (X, Y ) be a discrete bivariate random variable. Let

E X A M P L E S Example: Roll a pair of unbiased dice. If X denotes the smaller and Y denotes the larger outcome on the dice, then what is the joint probability density function of X and Y ?

MORE EXAMPLES A group of 9 executives of a certain firm include 4 who are married, 3 who never married, and 2 who are divorced. Three of the executives are to be selected for promotion. Let X denote the number of married executives and Y the number of never married executives among the 3 selected for promotion. Assuming that the three are randomly selected from the nine available, what is the joint probability density function of the random variables X and Y ?

PROBLEMS ON MARGINAL DISTRIBUTION Let X and Y be discrete random variables with joint probability density function

JOINT PROBABILITY DENSITY FUNCTION A real valued function f of two variables is a joint probability density function of a pair of discrete random variables X and Y (with range spaces RX and RY , respectively) if and only if For what value of the constant k the function given by

J O I N T C U M U L A T I V E P R O B A B I L I T Y D I S T R I B U T I O N

BIVARIATE CONTINUOUS RANDOM VARIABLE The joint probability density function of the random variables X and Y is an integrable function f(x, y) such that

Q U E S T I O N A R I E S A N D F E E D B A C K