Statistical Methods in Practice STAT/MATH 3379 Dr. A. B. W. Manage Associate Professor of Mathematics & Statistics Department of Mathematics & Statistics Sam Houston State University Overview 6.1 Discrete Random Variables 6.2 Binomial Probability Distribution 6.3 Continuous Random Variables and the Normal Probability Distribution 6.4 Standard Normal Distribution 6.5 Applications of the Normal Distribution 6.1 Discrete Random Variables Objectives: By the end of this section, I will be able to 1) Identify random variables. 2) Explain what a discrete probability distribution is and construct probability distribution tables and graphs. 3) Calculate the mean, variance, and standard deviation of a discrete random variable. 1
Random Variables A variable whose values are determined by chance Chance in the definition of a random variable is crucial Example: Notation for random variables Suppose our experiment is to toss a single fair die, and we are interested in the number rolled. We define our random variable X to be the outcome of a single die roll. a. Why is the variable X a random variable? b. What are the possible values that the random variable X can take? c. What is the notation used for rolling a 5? d. Use random variable notation to express the probability of rolling a 5. Example continued Main types of random variables a) We don t know the value of X before we toss the die, which introduces an element of chance into the experiment b) Possible values for X: 1, 2, 3, 4, 5, and 6. c) When a 5 is rolled, then X equals the outcome 5, or X = 5. Discrete random variable - a finite or a countable number of values Continuous random variable can take infinitely many values d) Probability of rolling a 5 for a fair die is 1/6, thus P(X = 5) = 1/6. 2
Example - Discrete and continuous random variables For the following random variables, (i) determine whether they are discrete or continuous, and (ii) indicate the possible values they can take. a. The number of automobiles owned by a family b. The width of your desk in this classroom c. The number of games played in the next World Series d. The weight of model year 2007 SUVs Example continued a. Since the possible number of automobiles owned by a family is finite and may be written as a list of numbers, it represents a discrete random variable. The possible values are {0, 1, 2, 3, 4,...}. Example continued Example 6.3 continued b. Width is something that must be measured, not counted. Width can take infinitely many different possible values, with these values forming an interval on the number line. Thus, the width of your desk is a continuous random variable. The possible values might be 1 ft W 10 ft. c. The number of games played in the next World Series can be counted and thus represents a discrete random variable. The possible values are {4, 5, 6, 7}. 3
Example continued d. The weight of model year 2007 SUVs must be measured, not counted, and so represents a continuous random variable. Weight can take infinitely many different possible values in an interval: possible values 2500 lb Y 7000 lb. Discrete Probability Distributions Provides all the possible values that the random variable can assume Together with the probability associated with each value Can take the form of a table, graph, or formula Describe populations, not samples Example - Probability distribution table Construct the probability distribution table of the number of heads observed when tossing a fair coin twice. Example continued The probability distribution table given in Table 6.2 uses probabilities we found in Example 6.4 page 261. Table 6.2 Probability distribution of number of heads on two fair coin tosses 4
Requirements Probability Distribution of a Discrete Random Variable The sum of the probabilities of all the possible values of a discrete random variable must equal 1. That is, ΣP(X) = 1. The probability of each value of X must be between 0 and 1, inclusive. That is, 0 P(X ) 1. Mean of a Discrete Random Variable The mean μ of a discrete random variable represents the mean result when the experiment is repeated an indefinitely large number of times Also called the expected value or expectation of the random variable X. Denoted as E(X ) Holds for discrete and continuous random variables Finding the Mean of a Discrete Random Variable Multiply each possible value of X by its probability. Add the resulting products. X P X Example: An auto insurance company pays $5000 to the customer in case of a loss due to theft. If the risk of a loss is assessed to be 3 in every 500 and the insurance company is selling this policy for $100, what is expected profit from each customer? 5
Variability of a Discrete Random Variable Formulas for the Variance and Standard Deviation of a Discrete Random Variable Definition Formulas 2 2 X P X 2 X P X Computational Formulas 2 2 2 X P X 2 2 X P X 6.2 Binomial Probability Distribution Objectives: By the end of this section, I will be able to 1) Explain what constitutes a binomial experiment. 2) Compute probabilities using the binomial probability formula. 3) Find probabilities using the binomial tables. 4) Calculate and interpret the mean, variance, and standard deviation of the binomial random variable. Binomial Experiment Four Requirements: 1) Each trial of the experiment has only two possible outcomes (success or failure) Notation Table 6.6 Notation for binomial experiments and the binomial distribution 2) Fixed number of trials 3) Experimental outcomes are independent of each other 4) Probability of observing a success remains the same from trial to trial 6
Formula for the Number of Combinations The number of combinations of X items chosen from n different items is given by n C X n! X! n X! Binomial Probability Distribution Formula The probability of observing exactly X successes in n trials of a binomial experiment is P(X) = ( n C X ) p X (1 - p) n-x where n! represents n factorial, which equals n(n - 1)(n - 2)... (2)(1); and 0! is defined to be 1. Binomial Distribution Tables n is the number of trials X is the number of successes p is the probability of observing a success Binomial Mean, Variance, and Standard Deviation Mean (or expected value): μ = n p Variance: σ 2 = n p (1 - p) Standard deviation: n p1 p FIGURE: Excerpt from the binomial tables. 7
Example - Mean, variance, and standard deviation of left-handed students Suppose we know that the population proportion p of left-handed students is 0.10. a. In a sample of 200 students, how many would we expect to be left-handed? b. Would 40 left-handed students out of 200 be considered unusual? Example 6.18 continued The binomial random variable here is X = the number of left-handed students. a. Here, n = 200, and p = 0.10. So the expected number of left-handed students in a sample of 200 is E(X) = μ = n p = (200)(0.10) = 20 Example continued b. To determine whether 40 lefties is unusual: 1. Find the standard deviation of X = the number of lefties in a sample of 200 students. 2. Calculate how many standard deviations 40 lies from the mean μ = 20. Example 6.18 continued The standard deviation of X is n p 1 p 2000.11 0.1 18 4.2426 8
Example continued How many standard deviations does 40 lie above the mean of 20? Simply take the difference between 40 and 20, and divide the result by the standard deviation 4.2426: X 40 20 4.714 4.2426 Finding 40 lefties in a sample of 200 is unusual because this value lies 4.7 standard deviations above the mean. 6.4 Continuous Random Variables and the Normal Probability Distribution Objectives: By the end of this section, I will be able to 1) Identify a continuous probability distribution. 2) Explain the properties of the normal probability distribution. FIGURE Figure continued (a) Relatively small sample (n = 100) with large class widths (0.5 lb). (c) Very large sample (n = 400) with very small class widths (0.1 lb). (b) Large sample (n = 200) with smaller class widths (0.2 lb). (d) Eventually, theoretical histogram of entire population becomes smooth curve with class widths arbitrarily small. 9
Continuous Probability Distributions A graph that indicates on the horizontal axis the range of values that the continuous random variable X can take Density curve is drawn above the horizontal axis Must follow the Requirements for the Probability Distribution of a Continuous Random Variable Requirements 1) The total area under the density curve must equal 1 (this is the Law of Total Probability for Continuous Random Variables). 2) The vertical height of the density curve can never be negative. That is, the density curve never goes below the horizontal axis. Probability Probability for Continuous Distributions is represented by area under the curve above an interval. The Normal Probability Distribution Most important probability distribution in the world Population is said to be normally distributed, the data values follow a normal probability distribution Specific population mean μ Specific population standard deviation σ μ and σ are parameters of the normal distribution 10
The normal distribution is symmetric about its mean μ. 6.5 Standard Normal Distribution Objectives: By the end of this section, I will be able to 1) Find areas under the standard normal curve, given a Z-value. 2) Find the standard normal Z-value, given an area. The Standard Normal (Z) Distribution A normal distribution with mean μ = 0 and standard deviation σ = 1. 6.6 Applications of the Normal Distribution Objectives: By the end of this section, I will be able to 1) Compute probabilities for a given value of any normal random variable. 2) Find the appropriate value of any normal random variable, given an area or probability. 3) Calculate binomial probabilities using the normal approximation to the binomial distribution. 11
Standardizing a Normal Random Variable Any normal random variable X can be transformed into the standard normal random variable Z by standardizing using the formula x Z 12