Chapter Objectives
1. Understand the difference between how probabilities are computed for discrete and continuous random variables.
2. Know how to compute probability values for a continuous uniform probability distribution and be able to compute the expected value and variance for such a distribution.
3. Be able to compute probabilities using a normal probability distribution. Understand the role of the standard normal distribution in this process.
4. Be able to use tables for the standard normal probability distribution to compute standard normal probabilities and probabilities for any normal distribution.
5. Be able to use Excel's NORMSDIST and NORMDIST functions to compute probability for the standard normal distribution and any normal distribution.
6. Be able to use Excel's NORMSINV and NORMINV function to find z and x values corresponding to given cumulative probabilities.
7. Be able to compute probabilities using an exponential probability distribution and understand Excel's EXPONDIST function.
8. Understand the relationship between the Poisson and exponential probability distribution.
1. Continuous Probability Distributions
• A continuous random variable can assume any value in an interval on the real line or in a collection of intervals.
• It is not possible to talk about the probability of the random variable assuming a particular value.
• Instead, we talk about the probability of the random variable assuming a value within a given interval.
• The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2.
• The total area under the graph f(x) is equal to 1.
2. Area as a Measure of Probability
• The area under the graph of f(x) and probability are the same.
• The probability that x takes a value between some lower value x1 and some higher value x2 can be found by computing the area under the graph of f(x) over the interval x1 and x2.
• Uniform Probability Distribution
o If a random variable is restricted to be within some interval and the probability density function is constant over the interval, (a,b), the continuous random variable is said to have a uniform distribution between a and b.
o Its density function is given by:
Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College
3. & 4. Normal Probability Distribution
• The normal distribution is perhaps the most widely used distribution for describing a continuous random variable.
• The normal probability density function is:
• Characteristics of the Normal Probability Distribution
o The shape of the normal curve is often illustrated as a bell-shaped curve.
o Two parameters, µ (mean) and σ (standard deviation), determine the location and shape of the distribution.
o The highest point on the normal curve is at the mean, which is also the median and mode.
o The mean can be any numerical value: negative, zero, or positive.
o The normal curve is symmetric (left and right halves are mirror images).
o The standard deviation determines the width of the curve: larger values result in wider, flatter curves.
o The total area under the curve is 1 (0.5 to the left of the mean and 0.5 to the right).
o Areas under the curve give probabilities for the normal random variable.
o Unlike the uniform distribution, the height of the normal distribution's curve varies and calculus is required to compute the areas that represent probability. .
o Areas under the normal curve have been computed and are available in tables that can be used in computing probabilities.
o The percentage of values in some commonly used intervals are:
• 68.26% of values of a normal random variable are within +/- 1 standard deviation of its mean.
• 95.44% of values of a normal random variable are within +/- 2 standard deviations of its mean.
• 99.72% of values of a normal random variable are within +/- 3 standard deviations of its mean.
• Standard Normal Probability Distribution
o A random variable that has a normal distribution with a mean of zero and a standard deviation of one is said to have a standard normal probability distribution.
o The letter z is commonly used to designate this normal random variable.
o We can think of z as a measure of the number of standard deviations x is from µ.
o The formula used to convert any normal random variable x, with mean µ and standard deviation
5. Using NORMDIST and NORMSDIST
• Using Excel’s NORMDIST Function
o The NORMDIST function computes the cumulative probability for any normal distribution. We do not need to convert an x value to a z value.
o The NORMDIST function requires four arguments
The x value for which we want to compute the cumulative probability
The mean
The standard deviation
A value of TRUE or FALSE (Normally we put TRUE because we want a cumulative probability)
o To compute a probability of a random variable assuming a value less than the x value, we use the function =NORMDIST. To computer a probability of a random variable assuming a value greater than the x value, we enter the formula =1-NORMDIST
• Using Excel’s NORMSDIST Function
o The NORMSDIST function is used to compute the cumulative probability for the standard normal distribution. Essentially, we are converting a normal distribution to the standard normal distribution by computing and using a z value rather than an x value.
o The NORMSDIST function requires one argument, which is the z value. To compute a probability of a random variable assuming a value less than the x value, we use the function =NORMSDIST. To computer a probability of a random variable assuming a value greater than the x value, we enter the formula =1-NORMSDIST .
6. Using NORMINV and NORMSINV
• Using Excel’s NORMINV Function
o The NORMINV function is used to compute the x value for a given cumulative probability for any normal distribution.
o The NORMINV function requires three arguments, which are the cumulative probability, the mean, and the standard deviation. To compute the x value, since we’re solving for it, we enter the formula =NORMINV into the cell.
• Using Excel’s NORMSINV Function
o The NORMSINV function is used to compute the x value for a given cumulative probability for the standard normal distribution.
o The NORMSINV function requires three arguments, which are the cumulative probability, the mean, and the standard deviation. To compute the x value, since we’re solving for it, we enter the formula =NORMSINV into the cell.
7. Exponential Probability Distribution
• A continuous probability distribution frequently used for computing the probability of the time to complete a task is the exponential distribution.
• If the average time to complete a task is denote by p, then the probability density function for the amount of time, x, to complete the task is given by:
Notes Compiled by Nicolas C. Rouse, Instructor, Phoenix College • Using Excel’s EXPONDIST Function
o Excel’s EXPONDIST function can be used to compute exponential probabilities.
o The EXPONDIST function has three arguments: the first is the value of x0, the second is the value of 1/µ, and the third is TRUE or FALSE. We will always select true because we are seeking a cumulative probability.
8. Relationship Between the Poisson and Exponential Distributions
• If the Poisson distribution provides an appropriate description of the number of occurrences per interval, the exponential distribution provides a description of the length of the interval between occurrences.
o As an example: If customers arrive according to a Poisson distribution with a mean of λ
customers, then the interarrival times of customers follows an exponential distribution with
µ = 1/
λ
Solving for an Upper “Tail” Area on a Normal Distribution
• Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed. The store manager is concerned that sales are being lost due to stockouts while waiting for a replenishment order. It has been determined that lead-time demand is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons.
• What is the probability of a stockout P(x > 20)?
o We are solving for the (tail) area to the right of the 20-gallon line in the graph below. The Cumulative Probabilities for the Standard Normal Distribution table in the textbook does not directly provide the area for the upper tail region of the distribution. So, we will first determine the area under the curve to the left of the reorder point.
We begin by converting our normal distribution (measured in gallons) to the standard normal distribution so that we can use the Standard Normal table of areas. Essentially, we must compute how many standard deviations lie between the mean demand value (15) and the reorder point value x (20).
z = (x - µ)/σ = (20 – 15)/6 = .83
The Cumulative Probabilities for the Standard Normal Distribution table shows an area of .7967 for the region to the left of the reorder point where z = .83.
KEY TERMS
Probability density function A function used to compute probabilities for a continuous
random variable. The area under the graph of a probability density function over an interval represents probability.
Uniform probability distribution A continuous probability distribution for which the
probability that the random variable will assume a value in any interval is the same for each interval of equal length.
Normal probability distribution A continuous probability distribution. Its probability
density function is bell-shaped and determined by its mean µ and standard deviation σ.
Standard normal probability distribution A normal distribution with a mean of zero and a standard
deviation of one.
Continuity correction factor A value of .5 that is added to or subtracted from a value of x
when the continuous normal distribution is used to approximate the discrete binomial distribution.
Exponential probability distribution A continuous probability distribution that is useful in computing
probabilities for the time it takes to complete a task.