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Chapter 9
Continuous Probability Models
9.1 Introduction
We have seen how discrete random variables can be modelled by discrete probability distribu- tions such as the binomial and Poisson distributions. We now consider how to model continuous random variables. A variable is discrete if it takes a countable number of values, for example, r = 0, 1 , 2 ,... , n or r = 0, 1 , 2 ,... or r = 0, 0. 1 , 0. 2 ,... , 0. 9 , 1. 0. In contrast, the values which a continuous variable can take form a continuous scale. One simple example of a continuous vari- able is height. Although in practice we might only record height to the nearest cm, if we could measure height exactly (to billions of decimal places) we would find that everyone had a different height. This is the essential difference between discrete and continuous variables. Therefore, if we could measure the exact height of every one of the n people on the planet, we would find that, for any height x, the proportion of people of height x is either 1 /n or 0. And if we imagine the number of people on the planet growing over time (n → ∞), this proportion tends to zero. This feature poses a problem for modelling continuous random variables as we can no longer use the methods we have seen work for discrete random variables.
The solution can be found by considering a (relative frequency) histogram of a sample of values taken by the continuous random variable, and thinking about what happens to the histogram as the sample size increases. For example, consider the following graphs which show histograms for samples of 100, 10000 and 1000000 observations made on a continuous random variable which can take values between 0 and 20. The final graph shows what happens when the sample size becomes infinitely big. This final graph is called the probability density function.
n=
x
Density
0 5 10 15 20
n=
x
Density
0 5 10 15 20
n=
x
Density
0 5 10 15 20
0 5 10 15 20
x
Density
Probability Density Function
As the population sizes gets larger, the histogram intervals get smaller and the jagged profile of the histogram smooths out to become a curve. We call this curve the probability density function (pdf) and it is usually written as f (x). Note that probabilities such as P (X < x) can be determined using the pdf as they equate to areas under the curve.
The key features of pdfs are
- pdfs never take negative values
- the area under a pdf is one: P (−∞ < X < ∞) = 1
- areas under the curve correspond to probabilities
- P (X ≤ x) = P (X < x) since P (X = x) = 0.
Therefore, for example, the probability that the inspectors visit the hotel in the morning (within 180 minutes after 9am) is
P (X ≤ 180) =
The probability of a visit during the lunch hour (12.30 to 13.30) is
P (210 ≤ X ≤ 270) = P (X ≤ 270) − P (X < 210)
=
9.2.1 Mean and Variance
The mean and variance of a continuous random variable can be calculated in a similar manner to that used for a discrete random variable. However the specific techniques required to do this are outside the scope of this course and so we will simply state the results.
If X is a uniform random variable on the interval a to b then its mean and variance are
E(X) = μ =
a + b 2
, V ar(X) = σ^2 =
(b − a)^2 12
In the above example, we have
E(X) =
a + b 2
so that the mean arrival of the inspectors is 9 am + 270 minutes = 13. 30. Also
V ar(X) =
(540 − 0)^2
and therefore SD(X) =
V ar(X) =
24300 = 155. 9 minutes.
9.3 The Exponential Distribution
The exponential distribution is another common distribution that is used to describe continuous random variables. It is often used to model lifetimes of products and times between “random” events such as arrivals of customers in a queueing system or arrivals of orders. The distribution has one parameter, λ. Its probability density function is
f (x) =
λe−λx^ for x ≥ 0 , 0 otherwise
and probabilities can be calculated using
P (X ≤ x) =
0 for x < 0 1 − e−λx^ for x > 0.
9.4 Exercises 9
- An express coach is due to arrive in Newcastle from London at 23.00. However in practice it is equally likely to arrive anywhere between 15 minutes early to 45 minutes late, depending on traffic conditions. Let the random variable X denote the amount of time (in minutes) that the coach is delayed.
(a) Sketch the pdf. (b) Calculate the mean and standard deviation of the delay time. (c) What is the probability that the coach is less than 5 minutes late? (d) What is the probability that the coach is more than 20 minutes late? (e) What is the probability that the coach arrives between 22.55 and 23.20? (f) What is the probability that the coach arrives at 23.00? (g) What is the probability that the coach arrives at 0.00? (h) Do you think that this is a good model for the coach’s arrival time?
- A network server receives incoming requests according to a Poisson process with rate λ =
- 5 per minute.
(a) What is the expectation of the time between arrivals of requests? (b) What is the probability that the time between requests is less than 2 minutes? (c) What is the probability that the time between requests is greater than 1 minute? (d) What is the probability that the time between requests is between 30 seconds and 50 seconds?
- As Production Manager, you are responsible for buying a new piece of equipment for your company’s production process. A salesman from one company has told you that he can supply you with equuipment for which the time to first breakdown (in months) follows an exponential distribution with λ = 0. 11. Another salesman (from another company) has told you that the time to first breakdown of their machines is also exponentially distributed but with λ = 0. 01. It is very important that the equipment you purchase does not break down for at least six months. Calculate the probability of this outcome for both suppliers and make a recommendation to the company board about which machine should be bought. How might you take into account a difference between the prices for the machines?
