Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 8. Probability Distributions: Part I Chapter 7 will be covered as a section of Chapter 8; This is the way I prefer to teach. Suppose someone offer you the following game (the Carnival Dice Game), will you be willing to put down $1 to play the game? You choose a number from {1,2,3,4,5,6}, then throw a die three times; record the number of times that your lucky number shows up. If your number shows up once, the host pays you $2; if the number shows up twice, you get $3; if the number shows up in all three times, you get $4. Otherwise, you lose your $1 bet. The Colorado Lotto and the Powerball To play , you pay $1 and select 6 numbers from 1 to 42. If you match all 6 numbers you hit the jackpot. There are three additional ways that you can win. The prize of jackpot starts from $1m and varies. To play , you pay $1 and select 5 numbers from 1 to 53 plus 1 number from 1 to 48. If you match these 5+1 numbers you hit the jackpot. There are eight additional ways that you can win. The prize of jackpot starts from $10m and varies. Should you play either lotto for a given jackpot prize? If you have only $1 and want to win big, should you play Colorado Lotto or the Powerball? Decision-Making and Probability Distributions Random Variables To make decision in a situation like those If the outcome of an experiment can assume mentioned above, one needs to know the odds (probability) for each possible outcome. A complete list of these probabilities is termed a probability distribution in statistics. With the probability distribution, a firm makes the decision among choices (scenarios) using the expected value. For an individual person, the criterion is the expected utility since people may be risk averse. at least two possible values and there is no certainty which value will be taken, then the outcome of the experiment is called a random variable. Examples. Non-random variables. There are two types of random variables: Discrete random variable: can assume only a finite number of values. Continuous random variable: can assume values over an entire interval or a range. 1 Probability Distribution of A Discrete Random Variable The Probability Distributions of the Three Lotteries We use a capital letter such as X for a random variable and x for all the possible values that X can assume. Throw a die once, X = the number on the die, x = 1, 2, 3, 4, 5, 6. Flip a coin 20 times, X = the number of heads, x = 0, 1, 2, …, 20. The probability distribution is the sequence of prob. P(X=x) for all x, or simply written as f(x). Throw a die once, f(x) = 1/6 for x = 1, 2, …, 6. We must have: 0 ≤ f(x) ≤1 and Σ f(x) = 1. More examples: Ex. 7.2 through 7.8 on p.162-3. Case Case Prize Prob. Match None $0 125/216 Match One $2 75/216 Match Two $3 Match Three $4 15/216 1/216 Case Prize Prob.:1 in Match 6/6 Jack 5245786 Match 5/6 $556 24287 Match 4/6 $46 556 Match 3/6 $4 37 The probability of not wining anything? 97.1% for the Lotto and 97.2% for the Powerball. c c 53 5 48 1 Prize Prob.: 1 in Match 5/5+1 J’pot 137744880 Match 5/5+0 $100k 2939678 Match 4/5+1 $5k 502195 Match 4/5+0 $100 12249 Match 3/5+1 $100 10685 Match 3/5+0 $7 261 Match 2/5+1 $7 697 Match 1/5+1 $4 124 Match 0/5+1 $3 80 c / 53 c5 48 c1 48 5 Mathematical Expectation and Variance of a Probability Distribution The Expectations of the Three Lotteries You may have observed that the number of people For the carnival dice game, its expectation or mean buying Powerball tickets increases as the amount of the jackpot gets larger. Why? This is because the expected value rises. For a random variable X, its mathematical expectation is µ ≡ EX = ∑ xf (x ) and its variance is σ X2 = ∑ ( x − µ )2 f ( x ) Both EX and σ are the same as Chapters 3&4. For f(x) = 1/6 with x ∈{1,…,6}, EX = 3.5 and σ = 1.71. is EX = ($0)(125/216) + ($2)(75/216) + ($3)(15/216) + ($4)(1/216) = $0.921. The meaning? For the Colorado Lotto with $1m jackpot, its mean is EX = ($1m)(1/5245786) + … = $0.213. Variance? For the Powerball with $10m jackpot, its mean is EX = ($10m)(1/120526770) + … = $0.173. Variance? All three games are called “unfair games” -- a fair game is the game with EX = the bet you put down. To make Lotto a fair game, the jackpot must be $4.12m; the jackpot for the Powerball is $99.7m. 2 The Binomial Probability Distribution In business and economics, we often encounter the following experiment: the so-called Bernoulli trial in which There is a fixed number of trials and all trial are independent; There are only two possible outcomes: success and failure; The probability of success (failure) remains the same through all trial. Cardinal dice game vs. Lotto Denote X the number of successes in n trials and p the probability of success in each trial, then the probability distribution is P ( X = x ) ≡ f ( x )= n C x p x (1 − p )n − x for x = 0,1, 2, …, n. The Binomial Probability Distribution If X assumes a binomial distribution, we denote it by X ∼ B(n,p). The use of Table V to find probabilities; the histogram of B(n,p) (Figure 8.2). The expectation, variance and standard deviation of a binomial distribution: µ ≡ EX = n ⋅ p 2 σ X = np(1 − p ) and σ X = np(1 − p ) Examples: Problem 8.46 and 8.48; Carnival dice game; telemarketing (10 & 100 calls). Other Probability Distributions The binomial Distribution is the most important discrete distribution, but there are others: The hypergeometric distribution: the trials are dependent; sampling with/without replacement; The Poisson distribution: if the number of trials is very large; The multinomial distribution: there are more than 2 possible outcomes in each trial. HW: 7.3 and 7.4; 8.4, 9, 12, and 14. Also 2 calculate EX, σ X and σ X of 8.5, 8.9 and 8.12. 3