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,