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Additional info for Fundamentals of Applied Probability Theory
We shall use I A) to denote the conditional compound PM 1: for random pr,l,lA(xO,~O variables x and givctn event A . 1 8 We begin by finding the conditional I'MF for random variable r, the length of the longest run obtained in three flips, given that the experimental value of h, the number of heads, is equal to 2. Thus, we wish to find p r l h ( r O ( 2). points in the original ho,ro INDEPENDENCE OF RANDOM VARIABLES 51 directly condifiori an event space by an arbitrary event defined on the experiment.
However, if we have an interest only in the behavior of random variable g and we wish' to answer several questions about it, we may desire t,owork in a go event space with the P D F f,(go). A P D F obtained for a function of some randoni variables whose P D F is known is referred to as a derived PDF. We shall introduce one simple method for obtaining a derived distribution by working in the event space of the random variables whose I'D17 is liuomn. There may be more efficient techniques for particular classes of problems.
Less axis, we may say that E ( x ) specifies the location of the c e n t e ~of mass of t,he PMF. d of the PMF for random variable x about its expected value. iance and is denoted by uZ2. 58 A conditional central moment is a measure of the nth power of t,he spread of a conditional PJIF for a random variable about its conditional mean. For instance, given that the experinwntal outconw had attribute A , the co~rditionalvariance of ratidom variable x, o:lA, is For functions of several random variables, me again define expect,at,ionto be the weighted sum of all possible experimental values of the function, with each such value weighted by the probability of its occurrence on a performance of the experiment.