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Unformatted text preview: 5 Joint mass and density functions The notions of discrete and continuous probability distributions may easily be extended to the case where we have two (or more) random variables. Suppose ( X,Y ) is a pair of random variables. The pair ( X,Y ) is said to be discrete if it can take on only a finite number of pairs of values. The probabilities with which ( X,Y ) is equal to each of those pairs are given by its joint probability mass function , or joint pmf, which we denote f XY . The probability that ( X,Y ) is equal to some fixed pair of values ( x,y ) is equal to f XY : P ( X = x and Y = y ) = f XY ( x,y ) . Since the values of f XY ( x,y ) for different ( x,y )’s represent probabilities, we must have f XY ( x,y ) ≥ 0 for all x and y , and the sum of the probabilities must equal one: ∑ x,y f XY ( x,y ) = 1. 5 The joint pmf of ( X,Y ) contains all information about the distribution of X and Y . In particular, we can extract f X and f Y , the pmfs for X and Y individually, from the joint pmf f XY . The individual pmfs f X and f Y are referred to as the marginal probability mass functions for X and Y . The marginal pmf of X evaluated at a point x is obtained by summing f XY ( x,y ) over all values of y , while the marginal pmf of Y evaluated at a point y is obtained by summing f XY ( x,y ) over all values of x : f X ( x ) = X y f XY ( x,y ) and f Y ( y ) = X x f XY ( x,y ) . A pair of random variables ( X,Y ) is said to be continuous if the probability of ( X,Y ) being equal to any fixed pair of values ( x,y ) is zero. In fact, if we want to be very precise, then continuity means a little more than this: the probability of ( X,Y ) lying in any subset of the plane that has zero area must be zero. When ( X,Y ) is continuous, its behavior may be described by its joint probability density function , or joint pdf, denoted f XY . If ( a,b ) and ( c,d ) are two intervals on the real line, then the probability that ( X,Y ) lies in the rectangle ( a,b ) × ( c,d ) – that is, the probability that X lies in ( a,b ) and Y lies in ( c,d ) – is equal to the volume underneath f X within the rectangle ( a,b ) × ( c,d ): P ( a < X < b and c < Y < d ) = Z d c Z b a f XY ( x,y )d x d y. Clearly, we must have f XY ( x,y ) ≥ 0 for all x and y , and the total volume under- neath the joint pdf must be one: R ∞-∞ R ∞-∞ f XY ( x,y )d x d y = 1. As in the discrete case, we may extract the pdfs f X and f Y for X and Y from the joint pdf f XY for ( X,Y ). The univariate pdfs f X and f Y are referred to as the marginal probability density functions , and may be obtained by “integrating out” one of the two arguments of f XY : f X ( x ) = Z ∞-∞ f XY ( x,y )d y and f Y ( y ) = Z ∞-∞ f XY ( x,y )d x....
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- Spring '08
- Normal Distribution, Probability theory, probability density function