05b STAT discrete random variables 2-LONG SLIDES

# 05b STAT discrete random variables 2-LONG SLIDES - DISCRETE...

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1 DISCRETE RANDOM VARIABLES Dr. V.R. Bencivenga Economic Statistics Economics 329 DISCRETE RANDOM VARIABLES PART 2 JOINT PROBABILITY DISTRIBUTIONS

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2 DISCRETE RANDOM VARIABLES Suppose we have two discrete random variables X and Y. Each is defined on the sample space S . When a basic outcome in S occurs, we see a pair of values, one for X, one for Y! k possible values of X k 2 1 x ..., , x , x h possible values of Y h 2 1 y ..., , y , y k h possible pairs of values ) y , x ( j i that X and Y jointly may take on!
3 DISCRETE RANDOM VARIABLES Suppose we have two discrete random variables X and Y. Each is defined on the sample space S . When a basic outcome in S occurs, we see a pair of values, one for X, one for Y! k possible values of X k 2 1 x ..., , x , x h possible values of Y h 2 1 y ..., , y , y k h possible pairs of values ) y , x ( j i that X and Y jointly may take on! Example #3 Stock portfolio We buy stock in two companies at the beginning of the year. At the end of the year, we have a rate of return on stock #1 (suppose it increased 5%, plus we got a dividend equal to 1% of what we paid for the stock). We also have a rate of return on stock #2 (suppose it increased 5%, no dividend). Random variables Realizations X = rate of return on stock #1 x = 6% Y = rate of return on stock #2 y = 5% One basic outcome occurs (stock market at the end of the year). A pair of values (x, y) is realized .

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4 DISCRETE RANDOM VARIABLES How can we describe the distribution of probability over possible pairs of values of X and Y? We use a joint probability function. Definition : The joint probability function of two discrete random variables X and Y gives the distribution of probability over possible pairs of values (x i , y j ): ) y Y x X ( P ) y , x ( P j i j i XY for k 2 1 x ..., , x , x and h 2 1 y ..., , y , y . ) y Y x X ( P ) y , x ( P j i j i XY is the probability of the intersection of two events: X = x i and Y = y j ! You give the probability function a pair of values, and it returns the probability the pair will occur!
5 DISCRETE RANDOM VARIABLES 1 y 2 y h y 1 x ) y , x ( P 1 1 ) y , x ( P 2 1 ) y , x ( P h 1 2 x ) y , x ( P 1 2 ) y , x ( P 2 2 k x ) y , x ( P 1 k ) y , x ( P h k We drop the subscript on P XY (x,y) when we can, to simplify notation. Definition : The joint probability distribution is the “list” of all possible pairs of values of X and Y, together with the probability that each pair will occur. This information may be reported using an equation, table, or graph.

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6 DISCRETE RANDOM VARIABLES We want to be able to analyze how the rate of return and the riskiness of a portfolio depend on the joint probability distribution and the “portfolio weights.” Example #3 Stock portfolio X = annual rate of return on stock #1 Y = annual rate of return on stock #2 X 4 5 6 7 5 .1 .05 .05 0 Y 6 .05 .15 .05 .05 7 .05 .05 .15 .25 in “ percentage points
7 DISCRETE RANDOM VARIABLES Properties of the probability function of a univariate discrete random variable: i. 0 ) x ( P i X ii. 1 ) x ( P i i X No negative probabilities.

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05b STAT discrete random variables 2-LONG SLIDES - DISCRETE...

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