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Unformatted text preview: 18 “mcs-ftl” — 2010/9/8 — 0:40 — page 467 — #473 Expectation 18.1 Definitions and Examples The expectation or expected value of a random variable is a single number that tells you a lot about the behavior of the variable. Roughly, the expectation is the average value of the random variable where each value is weighted according to its probability. Formally, the expected value (also known as the average or mean ) of a random variable is defined as follows. Definition 18.1.1. If R is a random variable defined on a sample space S , then the expectation of R is X Ex ŒR WWD R.w/ Pr Œw : (18.1) w 2 S For example, suppose S is the set of students in a class, and we select a student uniformly at random. Let R be the selected student’s exam score. Then Ex ŒR is just the class average—the first thing everyone wants to know after getting their test back! For similar reasons, the first thing you usually want to know about a random variable is its expected value. Let’s work through some examples. 18.1.1 The Expected Value of a Uniform Random Variable Let R be the value that comes up with you roll a fair 6-sided die. The the expected value of R is 1 1 1 1 1 1 7 Ex ŒR D 1 6 C 2 6 C 3 6 C 4 6 C 5 6 C 6 6 D 2 : This calculation shows that the name “expected value” is a little misleading; the random variable might never actually take on that value. You don’t ever expect to roll a 3 1 2 on an ordinary die! Also note that the mean of a random variable is not the same as the median . The median is the midpoint of a distribution. Definition 18.1.2. The median 1 of a random variable R is the value x 2 range .R/ 1 Some texts define the median to be the value of x 2 range .R/ for which Pr ŒR x < 1=2 and Pr ŒR > x 1=2 . The difference in definitions is not important. 468 “mcs-ftl” — 2010/9/8 — 0:40 — page 468 — #474 Chapter 18 Expectation such that 1 Pr ŒR x and 2 1 Pr ŒR > x < : 2 In this text, we will not devote much attention to the median. Rather, we will focus on the expected value, which is much more interesting and useful. Rolling a 6-sided die provides an example of a uniform random variable. In general, if R n is a random variable with a uniform distribution on f 1;2;:::;n g , then n Ex ŒR n D X i 1 D n.n C 1/ D n C 1 : n 2n 2 i D 1 18.1.2 The Expected Value of an Indicator Random Variable The expected value of an indicator random variable for an event is just the proba- bility of that event. Lemma 18.1.3. If I A is the indicator random variable for event A , then Ex ŒI A D Pr ŒA : Proof. Ex ŒI A D 1 Pr ŒI A D 1 C Pr ŒI A D D Pr ŒI A D 1 D Pr ŒA : (def of I A ) For example, if A is the event that a coin with bias p comes up heads, then Ex ŒI A D Pr ŒI A D 1 D p ....
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This note was uploaded on 01/19/2012 for the course CS 6.042J / 1 taught by Professor Tomleighton,dr.martenvandijk during the Fall '10 term at MIT.
- Fall '10
- Computer Science