CHAPTER 8 PROBABILITY DISTRIBUTIONS AND STATISTICS
8.1 Distribution of Random Variables
Random Variable
Probability Distribution of Random Variables
8.2 Expected Value
Mean Mean is the average value of a set of numbers.
Applied Example Find the average number of cars waiting in line at the bank s drive-in teller at the beginning of each 2-minute interval during the period in question.
Expected Value The following gives a general method for calculating the expected value (that is, the average or mean) of a random variable X.
Applied Example
8.3 Variance and Standard Deviation
Variance The mean, or expected value, of a random variable enables us to express an important property of the probability distribution associated with the random variable in terms of a single number. But the knowledge of the location, or central tendency, of a probability distribution alone is usually not enough to give a reasonably accurate picture of the probability distribution. Illustration. Olivia has ten packages of Brand A potato chips and ten packages of Brand B potato chips. After carefully measuring the weights of each package, she obtains the following results: We can verify that the mean weights for each of the two brands is 16 ounces. However, a cursory examination of the data now shows that the weights of the Brand B packages exhibit much greater dispersion about the mean than do those of Brand A. One measure of the degree of dispersion, or spread, of a probability distribution about its mean is given by the variance of the random variable associated with the probability distribution. A probability distribution with a small spread about its mean will have a small variance, whereas one with a larger spread will have a larger variance.
Variance of a Random Variable
Example
Standard Deviation
8.4 The Binomial Distribution
Bernoulli Trials An important class of experiments have (or may be viewed as having) two outcomes. Example: In a coin-tossing experiment, the two outcomes are heads and tails. An experiment in which a person is inoculated with a flu vaccine. Here, the vaccine may be classified as being effective or ineffective with respect to that particular person. Experiments with two outcomes are called Bernoulli trials or binomial trials. It is standard practice to label one of the outcomes of a binomial trial a success and the other a failure. Example: In a coin-tossing experiment, the outcome a head may be called a success, in which case the outcome a tail is called a failure.
Binomial Experiment A sequence of Bernoulli (binomial) trials is called a binomial experiment.
n, p, and q In a binomial experiment, it is customary to denote the number of trials by n, the probability of a success by p, and the probability of a failure by q. Because the event of a success and the event of a failure are complementary events, we have the relationship p + q = 1 or, equivalently, q = 1 p
Probabilities in Bernoulli Trials
Mean, Variance, and Standard Deviation of Binomial Distribution EXAMPLE 4 For the experiment in Examples, compute the mean, the variance, and the standard deviation of X
Applied Example
Finite and Continuous Probability Distributions The probability distributions discussed in the preceding sections were all associated with finite random variables that is, random variables that take on finitely many values. Such probability distributions are referred to as finite probability distributions. We also consider probability distributions associated with a continuous random variable that is, a random variable that may take on any value lying in an interval of real numbers. Such probability distributions are called continuous probability distributions.
Probability Density Functions A continuous probability distribution is defined by a function f whose domain coincides with the interval of values taken on by the random variable associated with the experiment. Such a function f is called the probability density function associated with the probability distribution, and it has the following properties: 1. f (x) is nonnegative for all values of x in its domain. 2. The area of the region between the graph of f and the x-axis is equal to 1
Normal Distributions Normal distributions are without a doubt the most important of all the probability distributions. Many phenomena such as the heights of people in a given population, the weights of newborn infants, the IQs of college students, the actual weights of 16-ounce packages of cereals, and so on have probability distributions that are normal. The normal distribution also provides us with an accurate approximation to the distributions of many random variables associated with random-sampling problems.
Normal Curve The graph of a normal distribution, which is bell shaped, is called a normal curve
Standard Normal Distribution The normal curve with mean = 0 and standard deviation = 1 is called the standard normal curve. The corresponding distribution is called the standard normal distribution. The random variable itself is called the standard normal random variable and is commonly denoted by Z.
Table of Standard Normal Distribution
Examples 1. Let Z be the standard normal variable. Find the values of a. P(Z < 1.24) b. P(Z > 0.5) c. P(0.24 < Z < 1.48) d. P(1.65 < Z < 2.02) 2. Let Z be the standard normal random variable. Find the value of z if z satisfies a. P(Z < z) =.9474 b. P(Z > z) =.9115 c. P(-z < Z < z) =.7888 3. Suppose X is a normal random variable with = 100 and = 20. Find the values of: a. P(X < 120) b. P(X > 70) c. P(75 < X < 110)
Applied Examples