Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Lecture 5: Mathematical Expectation Assist. Prof. Dr. Emel YAVUZ DUMAN MCB1007 Introduction to Probability and Statistics Istanbul˙ K¨ult¨ur University Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Outline 1 Introduction 2 The Expected Value of a Random Variable 3 Moments 4 Chebyshev’s Theorem 5 Moment Generating Functions 6 Product Moments 7 Conditional Expectation Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Outline 1 Introduction 2 The Expected Value of a Random Variable 3 Moments 4 Chebyshev’s Theorem 5 Moment Generating Functions 6 Product Moments 7 Conditional Expectation Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Introduction A very important concept in probability and statistics is that of the mathematical expectation, expected value, or briefly the expectation, of a random variable. Originally, the concept of a mathematical expectation arose in connection with games of chance, and in its simplest form it is the product of the amount a player stands to win and the probability that he or she will win. For instance, if we hold one of 10, 000 tickets in a raffle for which the grand prize is a trip worth $4, 800, our mathematical expectation is 1 4, 800 · =$0.48. 10, 000 This amount will have to be interpreted in the sense of an average - altogether the 10, 000 tickets pay $4, 800, or on the average $4,800 . 10,000 =$048 per ticket. Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Outline 1 Introduction 2 The Expected Value of a Random Variable 3 Moments 4 Chebyshev’s Theorem 5 Moment Generating Functions 6 Product Moments 7 Conditional Expectation Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product The Expected Value of a Random Variable Definition 1 If X is a discrete random variable and f (x) is the value of its probability distribution at x,theexpected value of X is E(X )= xf (x). x Correspondingly, if X is a continuous random variable and f (x)is the value of its probability density at x,theexpected value of X is ∞ E(X )= xf (x)dx. −∞ In this definition it is assumed, of course, that the sum or the integral exists; otherwise, the mathematical expectation is undefined. Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Example 2 Imagine a game in which, on any play, a player has a 20% chance of winning $3 and an 80% chance of losing $1. The probability distribution function of the random variable X , the amount won or lost on a single play is x $3 −$1 f (x) 0.20.8 and so the average amount won (actually lost, since it is negative) - in the long run - is E(X ) = ($3)(0.2) + (−$1)(0.8) = −$0.20. What does “in the long run” mean? If you play, are you guaranteed to lose no more than 20 cents? Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Solution. If you play and lose, you are guaranteed to lose $1. An expected loss of 20 cents means that if you played the game over and over and over and over ··· again, the average of your $3 winnings and your $1 losses would be a 20 cent loss. “In the long run” means that you can’t draw conclusions about one or two plays, but rather thousands and thousands of plays. Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Example 3 A lot of 12 television sets includes 2 with white cords. If three of the sets are chosen at random for shipment to a hotel, how many sets with white cords can the shipper expect to send to the hotel? Solution. Since x of the two sets with white cords and 3 − x of 2 10 the 10 other sets can be chosen in x 3−x ways, three of the 12 12 12 sets can be chosen in 3 ways, and these 3 possibilities are presumably equiprobable, we find that the probability distribution of X , the number of sets with white cords shipped to the hotel, is given by 2 10 x 3− x , , f (x)= 12 for x =0 1 2 3 Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product or, in tabular form, x 012 f (x) 6/11 9/22 1/22 Now, 2 6 9 1 1 E(X )= xf (x)=0· +1· +2· = 11 22 22 2 x=0 and since half a set cannot possibly be shipped, it should be clear that the term “expect” is not used in its colloquial sense. Indeed, it should be interpreted as an average pertaining to repeated shipments made under the given conditions. Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Example 4 A saleswoman has been offered a new job with a fixed salary of $290. Her records from her present job show that her weekly commissions have the following probabilities: commission 0 $100 $200 $300 $400 probability 0.05 0.15 0.25 0.45 0.1 Should she change jobs? Solution. The expected value (expected income) from her present job is (0 × $0.05) + (0.15 × $100) + (0.25 × $200) +(0.45 × $300) + (0.1 × $400) = $240. If she is making a decision only on the amount of money earned, then clearly she ought to change jobs. Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Example 5 Dan, Eric, and Nick decide to play a fun game. Each player puts $2 on the table and secretly writes down either heads or tails. Then one of them tosses a fair coin. The $6 on the table is divided evenly among the players who correctly predicted the outcome of the coin toss. If everyone guessed incorrectly, then everyone takes their money back. After many repetitions of this game, Dan has lost a lot of money - more than can be explained by bad luck. What’s going on? Solution. A tree diagram for this problem is worked out below, under the assumptions that everyone guesses correctly with probability 1/2 and everyone is correct independently. Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Nick right? Dan’s payoff Probability Eric right? Y $0 1/8 1/2 Y 1/2 / 1/2 N $1 1 8 Dan right? / 1 2 Y $1 1/8 Y N 1/2 1/2 1/2 N $4 1/8 Y −$2 1/8 1/2 1/2 N Y 1/2 − / 1/2 N $2 1 8 1/2 Y −$2 1/8 1/2 N 1/2 $0 1/8 N Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product In the payoff column, we are accounting for the fact that Dan has to put in $2 just to play. So, for example, if he guesses correctly and Eric and Nick are wrong, then he takes all $6 on the table, but his net profit is only $4. Working from the tree diagram, Dan’s expected payoff is: 1 1 1 1 1 1 E(payoff) =0 · +1· +1· +4· +(−2) · +(−2) · 8 8 8 8 8 8 1 1 +(−2) · +0· 8 8 =0. So the game perfectly fair! Over time, he should neither win nor lose money. A game is called fair if the expected value of the game is zero. Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Let say that Nick and Eric are collaborating; in particular, they always make opposite guesses. So our assumption everyone is correct independently is wrong; actually the events that Nick is correct and Eric is correct are mutually exclusive! As a result, Dan can never win all the money on the table. When he guesses correctly, he always has to split his winnings with someone else. This lowers his overall expectation, as the corrected tree diagram below shows: Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Nick right? Dan’s payoff Probability Eric right? Y $0 0 0 Y 1 / 1/2 N $1 1 4 Dan right? / 1 2 Y $1 1/4 Y N 1 1/2 0 N $4 0 Y −$2 0 0 1/2 N Y 1 − / 1/2 N $2 1 4 1/2 Y −$2 1/4 1 N 0 $0 0 N Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product From this revised tree diagram, we can work out Dan’s actual expected payoff 1 1 1 E(payoff) =0 · 0+1· +1· +4· 0+(−2) · 0+(−2) · 4 4 4 1 +(−2) · +0· 0 4 1 = − . 2 So he loses an average of a half-dollar per game! Introduction The Expected Value of a Random Variable Moments Chebyshev’s Theorem Moment Generating Functions Product Example 6 Consider the following game in which two players each spin a fair wheel in turn Player A Player B 4 5 2 12 6 The player that spins the higher number wins, and the loser has to pay the winner the difference of the two numbers (in dollars).