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Applied calculus on-line chapter: calculus applied to probability and statistics
Section 3. Exponential, Normal, and Beta Distributions

Exponential Density Function

You are an investment analyst, and you have information that mortgage lenders are failing continuously at 5% per year. What is the probability that a mortgage lender will fail sometime within the next x years?

To answer the question, suppose that you started with 100 mortgage lenders. Since they are failing continuously at a rate of 5% per year, the number left after x years is given by the decay equation


Thus, the percentage that will have failed by that time—and hence the probability that we are asking for—is given by

Now let X be the number of years a randomly chosen mortgage lender will take to fail. We have just calculated the probability that X is between 0 and x. In other words,

But we also know that

for a suitable probability density function. Thus,

The Fundamental Theorem of Calculus tells us that the derivative of the left side is f(x). Thus,

which is the probability density function we were seeking.

Question Does this function satisfy the mathematical conditions necessary for it to be a probability density function?

Answer First, the domain of f is [0, +\infty), since x refers to the number of years from now. Checking requirements (a) and (b) for a probability density function,

There is nothing special about the number 0.05. Any function of the form

with a a positive constant is a probability density function. A density function of this form is referred to as an exponential density function.

Exponential Density Function

An exponential density function is a function of the form

    f(x) = ae^{-ax} \qquad (a \text{ a positive constant})

with domain [0 +\infty). Its graph is shown in the figure.

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Example 1 Failing Mortgage Lenders

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Example 2 Radioactive Decay

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Normal Density Function

Perhaps the most interesting class of probability density functions are the normal density functions, defined as follows.

Normal Density Function

A normal density function is a function of the form

    f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{\Large{{-\ \frac{(x\ -\ \mu)^2}{2\sigma^2}}}},

with domain (-\infty, +\infty). The quantity \mu is called the mean and can be any real number, while \sigma is called the standard deviation and can be any positive real number. The graph of a normal density function is shown in the following figure:

Properties of the Normal Density Curve

You can check the following properties using calculus and a little algebra:

    (1) The normal curve is "bell-shaped" with the maximum occurring at x = \mu (center point marked on the graph).

    (2) It is symmetric about the vertical line x = \mu.

    (3) It is concave down in the range \mu-\sigma \leq x \leq \mu + \sigma, and concave up outside that range.

    (4) There are inflection points at x = \mu-\sigma and x = \mu-\sigma as marked on the graph.

    (5) (Less obvious) The integral of the normal density function is given in terms of the Gauss error function:

      \int f(x)\ dx = \frac{1}{2}\text{erf}\Bigl(\frac{x-\mu}{\sqrt{2}\sigma}\Bigr) + C

The normal density function applies in many situations that involve measurement and testing. For instance, repeated imprecise measurements of the length of a single object, a measurement made on many items from an assembly line, and collections of SAT scores tend to be distributed normally. It is for this reason that the normal density curve is so important in quality control and in assessing the results of standardized tests.

In order to use the normal density function to compute probabilities, we need to calculate integrals of the form \int_a^b f(x)\ dx. However, as we saw above, the antiderivative of the normal density function cannot be expressed in terms of any commonly used functions. Traditionally, statisticians and others have used tables coupled with transformation techniques to evaluate such integrals. This approach is rapidly becoming obsolete as the technology of hand-held computers and programmable calculators puts the ability to do numerical integration quickly and accurately in everybody's hands (literally). In keeping with this trend, we shall show how to use various technologies do the necessary calculation in the next example.

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Example 3 Quality Control    

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Beta Density Function

There are many random variables whose values are percentages or fractions. These variables have density functions defined on [0,1]. A large class of random variables, such as the percentage of new businesses that turn a profit in their first year, the percentage of banks that default in a given year, and the percentage of time a plant's machinery is inactive, can be modeled by a beta density function.

Beta Density Function

A beta density function is a function of the form

    f(x) = (\beta +1)( \beta +2)x^\beta(1 - x)

with domain [0, 1]. The number \beta can be any nonnegative constant. Below are the graphs of f(x) for several values of \beta. You can adjust the value of \beta in the last one (change the value and press "Return" or "Enter").

\beta = 0
f(x) = 2(1-x)
\beta = 0.5
f(x) = 3.75x^{0.5}(1-x)
\beta = 1
f(x) = 6x(1-x)
\beta = 3
f(x) = 20x^3(1-x)
\beta =  
f(x) = (\beta +1)( \beta +2)x^\beta(1 - x)

Example 4 Downsizing in the Utilities Industry

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Last Updated: April, 2008
Web materials copyright © 2008 Stefan Waner