Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

Chapter 5 Discrete Random Variables McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved. Discrete Random Variables 5.1 5.2 5.3 5.4 5.5 Two Types of Random Variables Discrete Probability Distributions The Binomial Distribution The Poisson Distribution (Optional) The Hypergeometric Distribution (Optional) 5.6 Joint Distributions and the Covariance (Optional) 5-2 LO5-1: Explain the difference between a discrete random variable and a continuous random variable. 5.1 Two Types of Random Variables Random variable: a variable that assumes numerical values that are determined by the outcome of an experiment ◦ Discrete ◦ Continuous Discrete random variable: Possible values can be counted or listed ◦ The number of defective units in a batch of 20 ◦ A listener rating (on a scale of 1 to 5) in an AccuRating music survey Continuous random variable: May assume any numerical value in one or more intervals ◦ The waiting time for a credit card authorization ◦ The interest rate charged on a business loan 5-3 LO5-2: Find a discrete probability distribution and compute its mean and standard deviation. 5.2 Discrete Probability Distributions The probability distribution of a discrete random variable is a table, graph or formula that gives the probability associated with each possible value that the variable can assume Notation: Denote the values of the random variable by x and the value’s associated probability by p(x) 5-4 LO5-2 Discrete Probability Distribution Properties 1. 2. For any value x of the random variable, p(x) 0 The probabilities of all the events in the sample space must sum to 1, that is… px 1 all x 5-5 LO5-2 Expected Value of a Discrete Random Variable The mean or expected value of a discrete random variable X is: m X x p x All x m is the value expected to occur in the long run and on average 5-6 LO5-2 Variance The variance is the average of the squared deviations of the different values of the random variable from the expected value The variance of a discrete random variable is: 2 X x m X p x 2 All x 5-7 LO5-2 Standard Deviation The standard deviation is the square root of the variance X 2 X The variance and standard deviation measure the spread of the values of the random variable from their expected value 5-8 LO5-3: Use the binomial distribution to compute probabilities. 5.3 The Binomial Distribution The binomial experiment characteristics… 1. Experiment consists of n identical trials 2. Each trial results in either “success” or “failure” 3. Probability of success, p, is constant from trial to trial The probability of failure, q, is 1 – p 4. Trials are independent If x is the total number of successes in n trials of a binomial experiment, then x is a binomial random variable 5-9 LO5-3 Binomial Distribution Continued For a binomial random variable x, the probability of x successes in n trials is given by the binomial distribution: n! x n- x px = p q x!n - x ! n! is read as “n factorial” and n! = n × (n-1) × (n-2) × ... × 1 0! =1 Not defined for negative numbers or fractions 5-10 LO5-3 Binomial Probability Table p = 0.1 P(x = 2) = 0.0486 Table 5.4 (a) for n = 4, with x = 2 and p = 0.1 5-11 LO5-3 Several Binomial Distributions Figure 5.7 5-12 LO5-3 Mean and Variance of a Binomial Random Variable If x is a binomial random variable with parameters n and p (so q = 1 – p), then Mean m = n•p Variance 2x = n•p•q Standard deviation x = square root n•p•q X npq 5-13 LO5-4: Use the Poisson distribution to compute probabilities (Optional). 5.4 The Poisson Distribution Consider the number of times an event occurs over an interval of time or space, and assume that 1. The probability of occurrence is the same for any intervals of equal length 2. The occurrence in any interval is independent of an occurrence in any non-overlapping interval If x = the number of occurrences in a specified interval, then x is a Poisson random variable 5-14 LO5-4 The Poisson Distribution Continued Suppose μ is the mean or expected number of occurrences during a specified interval The probability of x occurrences in the interval when μ are expected is described by the Poisson distribution px e m m x! x ◦ where x can take any of the values x = 0,1,2,3, … ◦ and e = 2.71828 (e is the base of the natural logs) 5-15 LO5-4 Poisson Probability Table μ = 0.4 P( x 3) Table 5.6 e 0.4 (0.4)3 0.0072 3! 5-16 LO5-4 Poisson Probability Calculations Table 5.7 5-17 LO5-4 Mean and Variance of a Poisson Random Variable If x is a Poisson random variable with parameter m, then Mean mx = m Variance 2x = m Standard deviation x is square root of variance 2x 5-18 LO5-4 Several Poisson Distributions Figure 5.10 5-19 LO5-5: Use the hypergeometric distribution to compute probabilities (Optional). 5.5 The Hypergometric Distribution (Optional) Population consists of N items ◦ r of these are successes ◦ (N-r) are failures If we randomly select n items without replacement, the probability that x of the n items will be successes is given by the hypergeometric probability formula r N r x n x P( x) N n 5-20 LO5-5 The Mean and Variance of a Hypergeometric Random Variable Mean r m x n N Variance 2 x r N n r n 1 N N N 1 5-21 LO5-5 Hypergeometric Example Population of six stocks Four have positive returns We randomly select three stocks Find P(x=2), mean, and variance r N r 4 2 x n x 2 1 62 P( x 2) 0.6 20 N 6 n 3 r 4 3 2 N 6 r N n 4 4 6 3 r n 1 3 1 0.4 N N N 1 6 6 6 1 m x n 2x 5-22 LO5-6: Compute and understand the covariance between two random variables (Optional). 5.6 Joint Distributions and the Covariance (Optional) 5-23 LO5-6 Calculating Covariance To calculate covariance, calculate: (x–μx)(y–μy)p(x,y) for each combination of x and y Example on prior slide yields –0.0318 A negative covariance says that as x increases, y tends to decrease in a linear fashion A positive covariance says that as x increases, y tends to increase in a linear fashion 5-24 LO5-6 Four Properties of Expected Values and Variances If a is a constant and x is a random variable, then μax=aμx 2. If x1,x2,…,xn are random variables, then μ(x1,x2,…,xn)= μx1 + μx2 + … + μxn 3. If a is a constant and x is a random variable, then σ2ax=a2σ2x 4. If x1,x2,…,xn are statistically independent random variables, then the covariance is zero 1. ◦ Also, σ2(x1,x2,…,xn)= σ2x1+ σ2x2+…+ σ2xn 5-25