cp ~bfarr/Probability_start.mws ~

You can copy the worksheet now, but you should read through the lab
before you load it into Maple. Once you have read to the exercises,
start up Maple, load
the worksheet `Probability_start.mws`, and go through it
carefully. Then you can start working on the exercises.

- The time it takes for a packet of information to travel from one location to another on the Internet.
- The number of miles that an automobile tire can be driven before it fails.
- The lengths of supposedly identical bolts manufactured by a particular production line.
- The speed of a particular gas molecule in a sample of a gas.

You may be more familiar with what are called discrete random variables, for example the number of heads obtained in ten tosses of a coin, which can only take a finite number of discrete values. In the case of a discrete random variable, the probability of a single outcome can be positive. For example, the probability that a single flip of a coin produces tails is 50%. The situation is very different when we consider a random variable like the number of miles a tire can be driven before failure, which can take any value from zero to something over miles. Since there are an infinite number of possible outcomes, the probability that the tire fails at exactly some number of miles, for example miles, is zero. However, we would expect that the probability that the tire would fail between miles and miles would not be zero, but would be a positive number.

A random variable that can take on a continuous range of values is called a continuous random variable. There turn out to be lots of applications of continuous random variables in science, engineering, and business, so a lot of effort has gone into devising mathematical models. These mathematical models are all based on the following definition.

- , for all
- where represents the probability that the random variable is greater than or equal to but less than or equal to .

For example, consider the following function.

This function is non-negative, and also satisfies the second condition, since

which is pretty easy to show. So this could be a probability density function for a continuous random variable .

A lot of the effort involved in modeling a random process, that is, a
process whose outcome is a random variable, is in finding a suitable
probability density function. Over the years, lots of different
functions have been proposed and used. One thing that they all have in
common, though, is that they depend on parameters. For example, the
general exponential probability density function is defined as

where is a parameter that can be adjusted to get the best fit to any particular situation. In the exercises, you will be asked to show that only positive values of make sense.

The process of deciding what probability density function to use and
how to determine the parameters is very complicated and can involve
very sophisticated mathematics. However, in the simple approach we are
taking here, the problem of determining the parameter value(s) often
depends on quantities that can be determined experimentally, for
example by collecting data on tire failure. For our purposes, the two
most important quantities are the mean, and the standard
deviation . The mean is defined by

and the standard deviation is the square root of the variance, , which is defined by

In practice, the variance is often computed as follows,

which can be easily be obtained by expanding and writing as the sum of three integrals.

Probably the most important distribution is the normal distribution,
widely referred to as the bell-shaped curve. The probability density
function for a normal distribution with mean and standard
deviation is given by the following equation.

This distribution has a tremendous number of applications in science, engineering, and business. The exercises provide a few simple ones.

In applications, one generally has to know in advance that the random variable you want to model has, approximately, a certain kind of distribution. How one would determine this is way beyond the scope of this course, so we won't really discuss it. On the other hand, once you know, for example, that your random variable has a normal distribution you only need the values of the mean and the standard deviation to be able to model it. The exponential distribution is even simpler, since it only has one parameter, and you only need to know the mean of your random variable to use this distribution to model it.

One thing to keep in mind when you are using the normal distribution
as a model is that calculations can involve values of your random
variable that don't make physical sense. For example, suppose that a
machining operation produces steel shafts whose diameters have
a normal distribution, with a mean of inches and a standard
deviation of inch. If you were asked to compute the percentage of the
shafts in a certain production run had diameters less than
inches you would use the following integral

even though negative values for the shaft diameters don't make sense.

- Show that the probability density function given for the
exponential distribution,

satisfies the condition

as long as is a positive number. What would happen if was negative? - Show that the mean and the standard deviation of the exponential
distribution are both equal to .
- The weekly rainfall totals for a section of the midwestern
United States follow an exponential distribution with a mean of 1.6
inches. Find the probability that the weekly rainfall total in this
section will exceed 2 inches.
- The median of a random variable is the smallest value of
the random
variable such that
. If has an exponential
distribution with mean , find the median .
- Compute the mean and the variance of the gamma distribution.
- Suppose that the winning bids (in dollars) for Elvis memorabilia
on eBay
approximately follow a gamma distribution with and .
- Find the mean and variance of these winning bids.
- Using the distribution, estimate the fraction of winning bids that are higher than $50.

- The time until first failure of a brand of laser printers is
approximately normally distributed with a mean of 5000 hours and a
standard deviation of 400 hours.
- What fraction of these printers will fail before 5000 hours? Does your answer make sense? Please explain.
- What should be the guarantee time for these printers if the manufacturer wants only 5 % to fail within the guarantee period?

2001-01-16