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Statistics 510: Notes 17 Reading: Sections 6.1-6.2 I. Joint distribution functions (Chapter 6.1) Thus far, we have focused on probability distributions for single random variables. However, we are often interested in probability statements concerning two or more random variables. The following examples are illustrative: In ecological studies, counts, modeled as random variables, of several species are often made. One species is often the prey of another; clearly, the number of predators will be related to the number of prey. The joint probability distribution of the x, y and z components of wind velocity can be experimentally measured in studies of atmospheric turbulence. The joint distribution of the values of various physiological variables in a population of patients is often of interest in medical studies. A model for the joint distribution of age and length in a population of fish can be used to estimate the age distribution from the length distribution. The age distribution is relevant to the setting of reasonable harvesting policies. The joint behavior of two random variables X and Y is determined by the joint cumulative distribution function (cdf): 1 FXY ( x, y) P( X x, Y y) , where X and Y are continuous or discrete. The probability that ( X , Y ) belongs to a given rectangle is P( x1 X x2 , y1 Y y2 ) F (x , y ) F (x , y ) F (x , y ) F (x , y ) . 2 2 2 1 1 2 1 1 In general, if X 1 , , X n are jointly distributed random variables, the joint cdf is F ( x1 , , xn ) P( X1 x1 , X 2 x2 , , X n xn ) . Two- and higher-dimensional versions of probability distribution functions and probability mass functions exist. We start with a detailed description of joint probability mass functions. Joint probability mass functions: Let X and Y be discrete random variables defined on the sample space that take on values x1 , x2 , and y1 , y2 , respectively. The joint probability mass function of ( X , Y ) is p( xi , y j ) P( X xi , Y y j ) . Example 1: A fair coin is tossed three times independently: let X denote the number of heads on the first toss and Y denote the total number of heads. Find the joint probability mass function of X and Y. The joint pmf is given in the following table: 2 y 0 1/8 0 x 0 1 1 2/8 1/8 2 1/8 2/8 3 0 1/8 Marginal probability mass functions: Suppose that we wish to find the pmf of Y from the joint pmf of X and Y. pY (0) P (Y 0) P (Y 0, X 0) P (Y 0, X 1) 1 0 8 1 8 pY (1) P (Y 1) P (Y 1, X 0) P (Y 1, X 1) 2 1 8 8 3 8 In general, to find the frequency function of Y, we simply sum down the appropriate column of the table giving the joint pmf of X and Y. For this reason, pY is called the marginal probability mass function of Y. Similarly, summing across the rows gives p X ( x) p( x, yi ) i 3 which is the marginal pmf of X. The case for several random variables is analogous. If X 1 , , X n are discrete random variables defined on the sample space, their joint pmf is p( x1 , , xn ) P( X1 x1 , , X n xn ) . The marginal pmf of X1 , for example, is pX1 ( x1 ) p( x1 , x2 , x2 , , xn , xn ) . Example 2: One of the most important joint distributions is the multinomial distribution which arises when a sequence of n independent and identical experiments is performed, where each experiment can result in any one of r possible outcomes, with respective probabilities p1 , , pr , p1 , , pr . If we let X i denote the number of the n experiments that result in outcome i, then n! P( X 1 n1 , , X r nr ) p1n1 prnr n1 ! nr ! r whenever n i 1 i n. Suppose that a fair die is rolled 9 times. What is the probability that 1 appears three times, 2 and 3 twice each, 4 and 5 once each and 6 not at all? 4 II. Jointly continuous random variables Joint PDF and Joint CDF: Suppose that X and Y are continuous random variables. The joint probability density function (pdf) of X and Y is the function f ( x, y ) such that for every set C of pairs of real numbers P(( X , Y ) C ) f ( x, y )dxdy ( x , y )C Another interpretation of the joint pdf is obtained as follows: P{a X a da, b Y b db} b db b a da a f ( x, y )dxdy f (a, b)dadb when da and db are small and f ( x, y ) is continuous at a , b . Hence f (a, b) is a measure of how likely it is that the random vector ( X , Y ) will be near (a, b) This is similar to the interpretation of the pdf f ( x ) for a single random variable X being a measure of how likely it is to be near x . The joint CDF of ( X , Y ) can be obtained as follows: 5 F (a, b) P{ X ( , a ], Y ( , b]} f ( x, y )dxdy X ( , a ],Y ( ,b ] b a f ( x, y )dxdy It follows upon differentiation that 2 f ( a, b) F ( a, b) ab wherever the partial derivatives are defined. Marginal PDF: The cdf and pdf of X can be obtained from the pdf of ( X , Y ) : P( X x) P{ X x, Y (, )} x f ( x, y)dydx d d x P( X x) f ( x , y ) dydx f ( x, y)dy dx dx Similarly, the pdf of Y is given by f X ( x) fY ( y) f ( x, y)dx Example 3: Consider the pdf for X and Y 12 f ( x, y ) ( x 2 xy ), 0 x 1, 0 y 1 7 (a) Find P ( X Y ) (b) Find the marginal density f X ( x) of X . 6 7 III. Independent Random Variables (Chapter 6.2) The random variables X and Y are said to be independent if for any two sets of real numbers A and B , P{ X A, Y B} P{ X A}P{Y B} . Loosely, speaking X and Y are independent if knowing the value of one of the random variables does not change the distribution of the other random variable (we will develop this interpretation more in Chapter 6.4-6.5). Random variables that are not independent are said to be dependent. For discrete random variables, the condition of independence is equivalent to p( x, y) pX ( x) pY ( y) for all x, y . For continuous random variables, the condition of independence is equivalent to f ( x, y) f X ( x) fY ( y) for all x, y . Example 4: Consider ( X , Y ) with joint pmf 1 p (10,1) p(20,1) p(20, 2) 10 1 3 p (10, 2) p(10,3) , and p(20,3) 5 10 Are X and Y independent? 8 Example 5: Suppose that a node in a communications network has the property that if two packets of information arrive within time of each other they “collide” and then have to be retransmitted. If the times of arrival of the two packets are independent and uniform on [0, T ] , what is the probability that they collide? 9