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PreMBA Analytical Methods
Continuous Probability Distributions: Continuous Random Variables

You probably recall from the Discrete Probability Distributions section of this course that a discrete random variable assumes values that are separate and distinct. The probability distribution for a discrete random variable, such as the number of orders taken in one week, may look like the following graph.

Alternatively, a continuous random variable can assume a range of values that falls along a continuum. The probability distribution for a continuous random variable can be represented by a curve that spans the range of values that the variable can assume. The curve is continuous and will not have distinct segments as in the discrete distribution above.

Consider the continuous probability distribution below, which illustrates the range of temperatures at which auto parts fractures occur. The distribution shows that fractures occur between approximately –45 and –15 degrees Fahrenheit.

A continuous probability distribution can be used to determine the probability of a variable falling between any two chosen values within the range of the distribution. The automobile manufacturer uses the graph below to analyze the probability of the temperature being between –35 and –30 degrees Fahrenheit when parts fractures occur, which is indicated by the shaded area under the curve.

Integral calculus is used to find this area and calculate the probability, but that is beyond the scope of this course.

Consider a second business example: Each week, a manufacturer of plastic products generates a varying amount of waste during production. In order to maintain production schedules and avoid problems of underutilized plant capacity, the manufacturer must have a sufficient amount of raw material at its facility to compensate for the waste generated during production.

The company's operations executive needs to know the amount of waste generated each week. He will use this information to predict the probability associated with various levels of waste. This will allow him to determine the correct level of raw materials to have on hand for a production run.

The operations executive uses past data to construct a probability density function of the amount of waste generated each week. This distribution is illustrated below.

Using this probability distribution, the executive can see that the amount of waste ranges from 0 to 2,200 pounds, with the most probable amount of waste being approximately 1,100 pounds.

If the executive wanted to determine the probability that the level of waste will be between 800 and 1,400 pounds, he could calculate the area under the curve between 800 and 1,400. This area under the curve that he wishes to calculate is indicated by the shaded portion in the graph below.

Notice that the two continuous probability distributions you've seen here have similar shapes. Both are approximations of a useful distribution called the normal distribution, which will be examined in detail in the Normal Distribution portion of this course.

1. Classify each of the following random variables as discrete or continuous.

a. the number of people waiting in line at a checkout counter

b. the size of an intake valve on a 1965 Ford Mustang

c. the color of a macaroni-and-cheese dinner

d. the time it takes a customer to choose a product

Solution 1

2. How is it possible to measure the probability of temperatures, weights, times, or levels of satisfaction if there are an infinite number of values that the variable can take on?

Solution 2

3. Is it possible for a continuous random variable to have a binomial distribution? Why or why not?

Solution 3

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