Some Basic Relationships of Probability

Given an event A, the complement of A is defined to be the event consisting of all sample points that are not in A. The complement of A is denoted by Ac. Figure 4.4 is a diagram, known as a Venn diagram, which illustrates the concept of a complement. The rectangular area represents the

Conditional Probability

Often, the probability of an event is influenced by whether a related event already occurred. Suppose we have an event A with probability P(A). If we obtain new informa­tion and learn that a related event, denoted by B, already occurred, we will want to take advantage of this information by calculating a new probability

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Bayes’ Theorem

In the discussion of conditional probability, we indicated that revising probabilities when new information is obtained is an important phase of probability analysis. Often, we begin the analysis with initial or prior probability estimates for specific events of interest. Then, from sources such as a sample, a special report, or a product test, we

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Random Variables

A random variable provides a means for describing experimental outcomes using numer- and its associated experimen- ical values. Random variables must assume numerical values. In effect, a random variable associates a numerical value with each possible experimental outcome. The particular numerical value of the random variable depends on the outcome of the experiment. A

Developing Discrete Probability Distributions

The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. For a discrete random variable x, a probability function, denoted by f(x), provides the probability for each value of the random variable. The classical, subjective, and relative frequency methods of assign­ing probabilities can be used

Expected Value and Variance

The expected value, or mean, of a random variable is a measure of the central location for the random variable. The formula for the expected value of a discrete random variable x follows. Both the notations E(x) and m are used to denote the expected value of a random variable. Equation (5.4) shows that

Bivariate Distributions, Covariance, and Financial Portfolios

A probability distribution involving two random variables is called a bivariate probability distribution. In discussing bivariate probability distributions, it is useful to think of a bivariate experiment. Each outcome for a bivariate experiment consists of two values, one for each random variable. For example, consider the bivariate experiment of rolling a pair of dice.

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Binomial Probability Distribution

The binomial probability distribution is a discrete probability distribution that has many applications. It is associated with a multiple-step experiment that we call the binomial experiment. 1. A Binomial Experiment A binomial experiment exhibits the following four properties. PROPERTIES OF A BINOMIAL EXPERIMENT The experiment consists of a sequence of n identical trials. Two

Poisson Probability Distribution

In this section we consider a discrete random variable that is often useful in estimating the number of occurrences over a specified interval of time or space. For example, the random variable of interest might be the number of arrivals at a car wash in one hour, the number of repairs needed in 10

Hypergeometric Probability Distribution

The hypergeometric probability distribution is closely related to the binomial distribution. The two probability distributions differ in two key ways. With the hypergeometric distribu­tion, the trials are not independent; and the probability of success changes from trial to trial. In the usual notation for the hypergeometric distribution, r denotes the number of elements in

Uniform Probability Distribution

Consider the random variable x representing the flight time of an airplane traveling from Chicago to New York. Suppose the flight time can be any value in the interval from 120 minutes to 140 minutes. Because the random variable x can assume any value in that interval, x is a continuous rather than a

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Normal Probability Distribution

The most commonly used probability distribution for describing a continuous random variable is the normal probability distribution. The normal distribution has been used in a wide variety of practical applications in which the random variables are heights and weights of people, test scores, scientific measurements, amounts of rainfall, and other similar values. It is

Normal Approximation of Binomial Probabilities

In Section 5.5 we presented the discrete binomial distribution. Recall that a binomial experi­ment consists of a sequence of n identical independent trials with each trial having two possible outcomes, a success or a failure. The probability of a success on a trial is the same for all trials and is denoted by p.

Exponential Probability Distribution

The exponential probability distribution may be used for random variables such as the time between arrivals at a hospital emergency room, the time required to load a truck, the distance between major defects in a highway, and so on. The exponential probability density function follows. As an example of the exponential distribution, suppose that

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The Electronics Associates Sampling Problem

The director of personnel for Electronics Associates, Inc. (EAI), has been assigned the task of developing a profile of the company’s 2500 managers. The characteristics to be identified include the mean annual salary for the managers and the proportion of managers having completed the company’s management training program. Using the 2500 managers as the

Selecting a Sample

In this section we describe how to select a sample. We first describe how to sample from a finite population and then describe how to select a sample from an infinite population. 1. Sampling from a Finite Population Statisticians recommend selecting a probability sample when sampling from a finite popu­lation because a probability sample

Point Estimation

Now that we have described how to select a simple random sample, let us return to the EAI problem. A simple random sample of 30 managers and the corresponding data on annual salary and management training program participation are as shown in Table 7.2. The notation x1, x2, and so on is used to

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Introduction to Sampling Distributions

In the preceding section we said that the sample mean X is the point estimator of the population mean m, and the sample proportion p is the point estimator of the population proportion p. For the simple random sample of 30 EAI managers shown in Table 7.2, the point estimate of m is X

Sampling Distribution of x

In the previous section we said that the sample mean X is a random variable and its probability distribution is called the sampling distribution of X. This section describes the properties of the sampling distribution of X. Just as with other probability distributions we studied, the sampling distribution of X has an expected value

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Sampling Distribution of p

The sample proportion p is the point estimator of the population proportion p. The formula for computing the sample proportion is where x = the number of elements in the sample that possess the characteristic of interest n = sample size As noted in Section 7.4, the sample proportion p is a random variable