mit6_042js10_chap20 random variable

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Chapter 20 Random Variables So far we focused on probabilities of events —that you win the Monty Hall game; that you have a rare medical condition, given that you tested positive; . . . . Now we focus on quantitative questions:  How many contestants must play the Monty Hall game until one of them nally wins? . . .  How long will this condition last?  How much will I lose playing 6.042 games all day? Random variables are the mathemat ical tool for addressing such questions. 20.1 Random V ariable Examples Denition 20.1.1. A random variable, R, on a probability space is a total function whose domain is the sample space. The codomain of R can be anything, but will usually be a subset of the real numbers. Notice that the name “rando m variable” is a misnomer; random vari ables are actually functions! For example, suppose we toss three independent, unbiased coins. Let C   be the number of heads that appear. Let M  = 1 if the three coins come up all heads or all tails, and let M  = 0 otherwise. Now every outcome of the three coin ips uniquely determines the values of C  and M . For example, if we ip heads, tails, heads, then C  = 2 and M  = 0. If we ip tails, tails, tails, then C  = 0 and M  = 1. In effect, C  counts the number of heads, and M  indicates whether all the coins match. Since each outcome uniquely determines C  and M , we can regard them as functions mapping outcomes to numb ers. For this experiment, the sample space is: S  = {HHH,HHT,HTH,HTT,THH,THT,TTH,TTT   } . Now C  is a function that maps each outcome in the sample space to a number as 459

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