2. Discrete Random Variables and Expectation

Transcription

1 2. Discrete Random Variables and Expectation In tossing two dice we are often interested in the sum of the dice rather than their separate values The sample space in tossing two dice consists of 36 events of equal probability, given by the ordered pairs of numbers {(1,1), (1,2),, (6, 6)} If the quantity we are interested in is the sum of the two dice, then we are interested in 11 events (of unequal probability) Any such function from the sample space to the real numbers is called a random variable MAT RandAl, Spring Jan Random Variables and Expectation Definition 2.1: A random variable (RV) on a sample space is a real-valued function on ; that is,. A discrete random variable is a RV that takes on only a finite or countably infinite number of values For a discrete RV and a real value, the event " includes all the basic events of the sample space in which assumes the value I.e., " represents the set ) = } MAT RandAl, Spring Jan

2 We denote the probability of that event by Pr = Pr, If is the RV representing the sum of the two dice, the event =4corresponds to the set of basic events {(1, 3), (2,2), (3, 1)} Hence, Pr = 4 = 3 36 = 1 12 MAT RandAl, Spring Jan Definition 2.2: Two RVs and are independent if and only if Pr(( )) = Pr Pr ) for all values and. Similarly, RVs,, are mutually independent if and only if, for any subset [1, ] and any values,, Pr ) = Pr MAT RandAl, Spring Jan

3 Definition 2.3: The expectation of a discrete RV, denoted by E], is given by = Pr where the summation is over all values in the range of. The expectation is finite if Pr, converges; otherwise, it is unbounded. E.g., the expectation of the RV representing the sum of two dice is = =7 MAT RandAl, Spring Jan As an example of where the expectation of a discrete RV is unbounded, consider a RV that takes on the value 2 with probability 12 for = 1,2, The expected value of is ] = = 1 expresses that ] is unbounded MAT RandAl, Spring Jan

4 Linearity of Expectations By this property, the expectation of the sum of RVs is equal to the sum of their expectations Theorem 2.1 [Linearity of Expectations]: For any finite collection of discrete RVs,, with finite expectations, = MAT RandAl, Spring Jan Proof: We prove the statement for two random variables and (general case by induction). The summations that follow are understood to be over the ranges of the corresponding RVs: = Pr ) = Pr ) + Pr ) = Pr ) + Pr ) = Pr + Pr ] + The first equality follows from Definition 1.2. In the penultimate equation uses Theorem 1.6, the law of total probability. MAT RandAl, Spring Jan

5 Let us now compute the expected sum of two standard dice Let, where represents the outcome of die for = 1,2 Then = 1 6 =7 2 Applying the linearity of expectations, we have =7 Linearity of expectations holds for any collection of RVs, even if they are not independent MAT RandAl, Spring Jan Lemma 2.2: For any constant and discrete RV, ]. Proof: The lemma is obvious for =0. For 0, ]= Pr Pr Pr. MAT RandAl, Spring Jan

6 Jensen's Inequality Let us choose the length of a side of a square uniformly at random from the range [1,99] What is the expected value of the area? We can write this as ] It is tempting to think of this as being equal to, but a simple calculation shows that this is not correct In fact, = = 50 = 2500 whereas = > 2500 MAT RandAl, Spring Jan More generally, [ ] ( ) Consider =() The RV is nonnegative and hence its expectation must also be nonnegative ] = [( ]) + ] ]+( ) ) To obtain the penultimate line, use the linearity of expectations To obtain the last line use Lemma 2.2 to simplify [[]] = [] [] MAT RandAl, Spring Jan

7 The fact that [ ] ( ) is an example of Jensen's inequality Jensen's inequality shows that, for any convex function, we have )] ]) Definition 2.4: A function is said to be convex if, for any and 1, + + (1 ) Lemma 2.3: If is twice differentiable function, then is convex if and only if "() 0 MAT RandAl, Spring Jan MAT RandAl, Spring Jan

8 Theorem 2.4 [Jensen's Inequality]: If is a convex function, then )] ]). Proof: We prove the theorem assuming that has a Taylor expansion. Let ]. By Taylor's theorem there is a value such that (x)+ ) 2 (x ) since ) > 0 by convexity. Taking expectations and applying linearity of and Lemma 2.2 yields: ] + )( ) = ]). MAT RandAl, Spring Jan The Bernoulli and Binomial Random Variables We run an experiment that succeeds with probability and fails with probability Let be a RV such that = iftheexperimentsucceeds, otherwise The variable is called a Bernoulli or an indicator random variable. Note that, for a Bernoulli RV, ]=1+(10==Pr(=1) MAT RandAl, Spring Jan

9 If we, e.g., flip a fair coin and consider heads a success, then the expected value of the corresponding indicator RV is 1/2 Consider a sequence of independent coin flips What is the distribution of the number of heads in the entire sequence? More generally, consider a sequence of independent experiments, each of which succeeds with probability If we let represent the number of successes in the experiments, then has a binomial distribution MAT RandAl, Spring Jan Definition 2.5: A binomial RV with parameters and, denoted by ), is defined by the following probability distribution on = 0,1,2,,: Pr = I.e., the binomial RV (BRV) equals when there are exactly successes and failures in independent experiments, each of which is successful with probability Definition 2.5 ensures that the BRV is a valid probability function (Definition 1.2): Pr = = 1 MAT RandAl, Spring Jan

10 We want to gather data about the packets going through a router We want to know the approximate fraction of packets from a certain source / of a certain type We store a random subset or sample of the packets for later analysis Each packet is stored with probability and packets go through the router each day, the number of sampled packets each day is a BRV with parameters and To know how much memory is necessary for such a sample, determine the expectation of MAT RandAl, Spring Jan If is a BRV with parameters and, then is the number of successes in trials, where each trial is successful with probability Define a set of indicator RVs,,, where =1if the th trial is successful and 0 otherwise Clearly, ]=and = and so, by the linearity of expectations, = MAT RandAl, Spring Jan

11 2.3. Conditional Expectation Definition 2.6: ] = Pr where the summation is over all in the range of The conditional expectation of a RV is, like, a weighted sum of the values it assumes Now each value is weighted by the conditional probability that the variable assumes that value MAT RandAl, Spring Jan-15 95, Suppose that we independently roll two standard six-sided dice Let be the number that shows on the first die, the number on the second die, and the sum of the numbers on the two dice Then = 2 = Pr =2 1 = MAT RandAl, Spring Jan

12 As another example, consider =5: = 5 = Pr =5 = Pr =5 Pr =5 = 136 = MAT RandAl, Spring Jan Lemma 2.5: For any RVs and, ]= Pr ], where the sum is over all values in the range of and all of the expectations exist. Proof: Pr =Pr = Pr Pr Pr = Pr = Pr ] MAT RandAl, Spring Jan

13 The linearity of expectations also extends to conditional expectations Lemma 2.6: For any finite collection of discrete RVs,, with finite expectations and for any RV, = ] MAT RandAl, Spring Jan Confusingly, the conditional expectation is also used to refer to the following RV Definition 2.7: The expression ] is a RV ) that takes on the value ] when ] is not a real value; it is actually a function of the RV Hence ] is itself a function from the sample space to the real numbers and can therefore be thought of as a RV MAT RandAl, Spring Jan

14 In the previous example of rolling two dice, =Pr = We see that is a RV whose value depends on If ] is a RV, then it makes sense to consider its expectation ] We found that + 72 Thus, = = 7 = ] MAT RandAl, Spring Jan More generally, Theorem 2.7: Y = ] Proof: From Definition 2.7 we have, where takes on the value when. Hence = Pr The right-hand side equals Y by Lemma 2.5. MAT RandAl, Spring Jan

15 Consider a program that includes one call to a process Assume that each call to process recursively spawns new copies of the process, where the number of new copies is a BRV with parameters and We assume that these random variables are independent for each call to What is the expected number of copies of the process generated by the program? MAT RandAl, Spring Jan To analyze this recursive spawning process, we use generations The initial process is in generation 0 Otherwise, we say that a process is in generation if it was spawned by another process in generation 1 Let denote the number of processes in generation Since we know that =1, the number of processes in generation 1 has a binomial distribution Thus, = MAT RandAl, Spring Jan

16 Similarly, suppose we knew that the number of processes in generation 1was, so Then Applying Theorem 2.7, we can compute the expected size of the th generation inductively We have ] By induction on, and using the fact that =1, we then obtain = MAT RandAl, Spring Jan The expected total number of copies of process generated by the program is given by = = If 1then the expectation is unbounded; if <1, the expectation is 1 (1 ) The # of processes generated by the program is bounded iff the # of processes spawned by each process is less than 1 This is a simple example of a branching process, a probabilistic paradigm extensively studied in probability theory MAT RandAl, Spring Jan

17 2.4. The Geometric Distribution Let us flip a coin until it lands onheads What is the distribution of the number of flips? This is an example of a geometric distribution It arises when we perform a sequence of independent trials until the first success, where each trial succeeds with probability Definition 2.8: A geometric RV with parameter is given by the following probability distribution on = 1,2, : Pr ) = MAT RandAl, Spring Jan Geometric RVs are said to be memoryless because the probability that you will reach your first success trials from now is independent of the number of failures you have experienced Informally, one can ignore past failures they do not change the distribution of the number of future trials until first success Formally, we have the following Lemma 2.8: For a geometric RV with parameter and for >0, Pr ) = Pr MAT RandAl, Spring Jan

18 When a RV takes values in the set of natural numbers = {0,1,2,3, } there is an alternative formula for calculating its expectation Lemma 2.9: Let be a discrete RV that takes on only nonnegative integer values. Then Proof: ]= Pr = Pr Pr = Pr = Pr ] MAT RandAl, Spring Jan For a geometric RV with parameter, Pr = = Hence = = 1 (1) = 1 Thus, for a fair coin where = 1/2, on average it takes two flips to see the first heads MAT RandAl, Spring Jan

19 Finding the expectation of a geometric RV with parameter using conditional expectations and the memoryless property of geometric RVs Recall that corresponds to the number of flips until the first heads given that each flip isheads with probability Let =0if the first flip istails and =1if the first flip isheads By the identity from Lemma 2.5, = Pr = 0 =0 + Pr = 1 = 1] = (1 )[ = 0] + [ = 1] MAT RandAl, Spring Jan If = 1 then = 1, so [ = 1] = 1 If =0, then >1 In this case, let the number of remaining flips (after the first flip until the first heads) be Then, by the linearity of expectations, ]=(1+1]+1=(1]+1 By the memoryless property of geometric RVs, is also a geometric RV with parameter Hence ] = ], since they both have the same distribution We therefore have ]=(1]+1= (1)[]+1, which yields [] = 1/ MAT RandAl, Spring Jan

20 Example: Coupon Collector's Problem Each box of cereal contains one of different coupons Once you obtain one of every type of coupon, you can send in for a prize Coupon in each box is chosen independently and uniformly at random from the possibilities and that you do not collaborate to collect coupons How many boxes of cereal must you buy before you obtain at least one of every type of coupon? MAT RandAl, Spring Jan Let be the number of boxes bought until at least one of every type of coupon is obtained If is the number of boxes bought while you had exactly 1different coupons, then clearly = The advantage of breaking into a sum of random variables, = 1,,, is that each is a geometric RV When exactly 1coupons have been found, the probability of obtaining a new coupon is =1 1 MAT RandAl, Spring Jan

21 Hence, is a geometric RV with parameter : = 1 = +1 Using the linearity of expectations, we have that = = +1 1 MAT RandAl, Spring Jan The summation harmonic number ) is known as the Lemma 2.10: The harmonic number = satisfies ) = ln (1). Thus, for the coupon collector's problem, the expected number of random coupons required to obtain all coupons is ln MAT RandAl, Spring Jan

22 Given the first and second moments, one can compute the variance and standard deviation of the RV Intuitively, the variance and standard deviation offer a measure of how far the RV is likely to be from its expectation Definition 3.2: The variance of a RV is ] The standard deviation of a RV is ]= MAT RandAl, Spring Jan The two forms of the variance in the definition are equivalent, as is easily seen by using the linearity of expectations Keeping in mind that is a constant, we have [ ]= ] ] MAT RandAl, Spring Jan