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MIT6_042JF10_lec17

MIT6_042JF10_lec17 - 17 “mcs-ftl” — 2010/9/8 — 0:40...

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Unformatted text preview: 17 “mcs-ftl” — 2010/9/8 — 0:40 — page 445 — #451 Random Variables and Distributions Thus far, we have focused on probabilities of events. For example, we computed the probability that you win the Monty Hall game, or that you have a rare medical condition given that you tested positive. But, in many cases we would like to more more. For example, how many contestants must play the Monty Hall game until one of them finally wins? How long will this condition last? How much will I lose gambling with strange dice all night? To answer such questions, we need to work with random variables. 17.1 Definitions and Examples Definition 17.1.1. 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 “random variable” is a misnomer; random variables are actually functions! For example, suppose we toss three independent 1 , unbiased coins. Let C be the number of heads that appear. Let M D 1 if the three coins come up all heads or all tails, and let M D otherwise. Every outcome of the three coin flips uniquely determines the values of C and M . For example, if we flip heads, tails, heads, then C D 2 and M D . If we flip tails, tails, tails, then C D and M D 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 func- tions mapping outcomes to numbers. For this experiment, the sample space is S D f HHH;HHT;HTH;HT T;THH;THT;T TH;T T T g and C is a function that maps each outcome in the sample space to a number as 1 Going forward, when we talk about flipping independent coins, we will assume that they are mutually independent. 446 “mcs-ftl” — 2010/9/8 — 0:40 — page 446 — #452 Chapter 17 Random Variables and Distributions follows: C.HHH/ D 3 C.THH/ D 2 C.HHT / D 2 C.THT / D 1 C.HTH/ D 2 C.T TH/ D 1 C.HT T / D 1 C.T T T / D 0: Similarly, M is a function mapping each outcome another way: M.HHH/ D 1 M.THH/ D M.HHT / D M.THT / D M.HTH/ D M.T TH/ D M.HT T / D M.T T T / D 1: So C and M are random variables. 17.1.1 Indicator Random Variables An indicator random variable is a random variable that maps every outcome to either 0 or 1. Indicator random variables are also called Bernoulli variables . The random variable M is an example. If all three coins match, then M D 1 ; otherwise, M D . Indicator random variables are closely related to events. In particular, an in- dicator random variable partitions the sample space into those outcomes mapped to 1 and those outcomes mapped to 0. For example, the indicator M partitions the sample space into two blocks as follows: HHH T T T HHT HTH HT T THH THT T TH : „ ƒ‚ … „ ƒ‚ … M D 1 M D In the same way, an event E partitions the sample space into those outcomes in E and those not in E . So E is naturally associated with an indicator random variable,...
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MIT6_042JF10_lec17 - 17 “mcs-ftl” — 2010/9/8 — 0:40...

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