1 Discrete random variables discretervs tex draft October 6 2009 1 1 What is a discrete random variable A random variable is a discrete random variable if it is a variable that can take only a countable number of values Or to use the words of MGB A random variable X will be de ned to be discrete if the range of X is countable I will use N to denote the countable number of values X is the rv and depending on the context use x or xi to denote a speci c value of X Not that it is not correct to say that the a discrete rv can take only a nite number of values Consider for example the density function f x N1 where x 1 2 3 N This is a di erent density function for every value of N no matter how large This function can have mass at an in nite number of integers The probability density function for a discrete density function speci es which points have positive probability mass and what those probabilities are Note that the value of f x is the probability that X x that is f x Pr x This was not the case for continuous rvs for a continuous rv f x is not a probability If X is a discrete rv we call f x a discrete density function or a probability density function or a probability function The probability density function is set of spikes vertical lines on the real line a spike at each value of X that has positive probability The height of the spike at X x is the probability that X x The function takes the value of zero at all other points insert example graph here f xi Pr xi 0 for all possible values of X So it is required that N X Pr xi 1 i 1 since the probability of something happening is one Not that this implies that Pr xi 1 1 With discrete random variables integration is replaced by summation Of course we can speak loosely and think of summization as a special type of integration and then use the term integration to refer to both intergration and summation One can always speak loosely Of course With discrete distributions there is a probability associated with each point in the real line not like with continuous rvs but most of these probabilities will be zero 2 MGB formally de nes a discrete density function as De nition 1 Any function f with domain the real line and counterdomain 0 1 is de ned to be a discrete density function if for some countable set x1 x2 xn f xj 0 for j 1 2 f x 0 for x 6 xj j 1 2 PN j 1 f xj 1 where the summation is over the points x1 x2 xn 3 1 2 The expected value and variance of a discrete rv in terms of f E X N X xi Pr xi i 1 N X xi f xi i 1 and var X N X xi E x 2 Pr xi xi E x 2 f xi i 1 N X i 1 Further note that E g X N X g xi Pr xi i 1 N X g xi f xi i 1 These are the discrete analogs of corresponding expectation functions for continuous rv s 4 1 3 1 3 1 Some examples of discrete density functions The Bernoulli distribution This is Jakob Bernoulli creator of the Bernoulli distribution there was a shitload of famous Bernoulli mathematicans and they did not all get along a both functional and dysfunctional Swiss family Consider a random variable x that can take one of two values zero and one such that P 1 p and P 0 1 p where 0 p 1 Is x a continuous or a discrete random variable Discrete The density function for x is P x fX x fX x p px 1 p 1 x for x 0 or 1 0 otherwise or written di erently f 1 p f 0 1 p f x 0 if x does not equal 0 or 1 Convince me that this is a legitimate density function Since 0 p 1 p and 1 p are both positive the function is never negative In addition fX 0 fX 1 1 p p 1 The Bernoulli is relevant whenever there are two alternatives and the probabilities of the two alternatives are p and 1 p What is E x E x 1 X xfX x 0 1 x 0 5 p 1 p p What is var x var x E x E x 2 1 X x E x 2 fX x x 0 1 X x p 2 fX x 0 p 2 1 p 1 p 2 p x 0 p2 1 p 1 p 1 p 2p p2 p p2 p3 p 2p2 p3 Three examples of a random variable that would have a Bernoulli distribution which side of a fair coin results from ipping a coin p 5 whether a queen 4 is drawn when a card is randomly drawn from a deck p 52 The parameter p could be the probability that one votes for McCain In which case 1 p is the probability that one votes for someone else Obama Ralph Imagine you cannot decide how to vote so make up your mind in the following fashion You arrive at the polling station with an urn that contains 62 red balls and 38 blue balls so p 62 In the booth you randomly draw a ball from your urn and that determines your vote 6 An aside the odds When there are two alternatives x 1 and x 0 with Pr x 1 p a Bernoulli it is common to interpret x 1 as a sucess In such common circumstances statisticans usually not economists like to talk about the odds of a sucess rather than the probability of a sucess where the odds O are de ned as p O 1 p O 8 6 4 2 0 2 0 4 0 6 0 8 p The odds p 1 1 1 1 0 111 11 5 1 5 1 0 6 1 6 p as a function of p 1 5 7 7 1 7 2 333 3 Note that 0 O p O 1 O and O is unde ned for p 1 Solving O p 1 p for p p 0 8 0 6 0 4 0 2 0 0 5 10 15 20 O p as a function of the odds O So talking in terms of the odds and talking in terms of p are two di erent ways of saying the same thing p O Note the the odds of a sucess is 1 if p 5 less than one if p 5 and greater than 1 if p 5 Think about the odds of your horse winning the race a sucess 1 3 2 Add your own examples here 8 1 3 3 Assume the rv X has …