Chapter 7 Continuous Probability Distributions

A continuous random variable has an infinite number of values within a particular range.

This chapter shows how to use three continuous probability distributions: the uniform probability distribution, the normal probability distribution, and the exponential probability distribution.

The Family of Uniform Probability Distributions

The uniform probability distribution is the simplest distribution for a continuous random variable. The distribution is rectangular in shape and is completely defined by its minimum and maximum values.

The Family of Normal Probability Distributions

A normal distribution is defined by its mean and standard deviation.

Characteristics of a Normal Distribution

1. Bell-shaped and has a single peak at the center of the distribution. The arithmetic mean, median, and mode are equal and located in the center of the distribution. The total area under the curve is 1.00. Half the area under the normal curve is to the right of this center point and the other half, to the left of it.
2. Symmetrical about the mean. If we cut the normal curve vertically at the center value, the shapes of the curves will be mirror images. Also, the area of each half is 0.5.
3. It falls off smoothly in either direction from the central value. That is, the distribution is asymptotic: the curve gets closer and closer to the X-axis but never actually touches it.
4. The location of a normal distribution is determined by the mean, µ. The dispersion or spread of the distribution is determined by the standard deviation, \(\sigma\).

There is not just one normal probability distribution, but rather a “family” of them.

Recall that discrete probability distributions show the specific likelihood a discrete value will occur. However, in a continuous probability distribution, areas below the curve define probabilities.

The Standard Normal Probability Distribution

The number of normal distributions is unlimited, each having a different mean (µ), standard deviation (\(\sigma\)) or both. The standard normal probability distribution can be used to determine the probabilities for all normal probability distributions. It is unique because it has a mean of 0 and a standard deviation of 1.

Any normal probability distribution can be converted into a standard normal probabililty distribution by subtracting the mean from each observation and dividing this difference by the standard deviation. The results are called z values or z scores.

z Value The signed distance between a selected value, designated x, and the mean, µ, divided by the standard deviation, \(\sigma\).

or

a z value is the distance from the mean, measured in units of the standard deviation.

Applications of the Standard Normal Distribution

The standard normal distribution is very useful for determining probabilities for any normally distributed random variable. The basic procedure is to find the z value for a particular value of the random variable based on the mean and standard deviation of its distribution. Then, using the z value, we can use the standard normal distribution to find various probabilities.

z values can be used to find areas under the normal curve. In a general sense this can be used with any probability distribution with the condition that it match the data you are working with. For example, if we wanted to determine what percentage of inmates recidivate between 2 and 3 years, we can use the corresponding z, t, etc. values to determine probabilities (percentages) under the curve.

To summarize, there are four situations for finding the area under the standard normal probability distribution.

1. To find the area between 0 and z or (-z), look up the probability directly in the table.
2. To find the area beyond z or (-z), locate the probability of z in the table and subtrat that probability from .500.
3. To find the area between two points on different sides of the mean, determine the z values and add the corresponding probabilities.
4. To find the area between two points on the same side of the mean, determine the z values and subtract the smaller probability from the larger.

The Family of Exponential Distributions

The exponential continuous probability distribution usually describes times between events in a sequence. The actions occur independently at a constant rate per unit of time or length. Because time is never negative, an exponential random variable is always positive.

Typical situations described by exponential distributions

• The service time for customers at the information desk of the Dallas Public Library.
• The time between “hits” on a website.
• The lifetime of a kitchen appliance.
• The time until the next phone call arives in a customer service center.

Features of the exponential distribution

• The exponential probability distribution is positively skewed.
• The distribution is defined by only one parameter known \(\lambda\) (lambda).
• \(\lambda\) is often referred to as the “rate” parameter.