1. In an experiment, if a mouse is administered dosage level A of a certain (harmless) hormone then there is a 0:2 probability that the mouse will show signs of aggression within one minute. For dosage levels B and C; the probabilities are 0:5 and 0:8; respectively. Ten mice are given exactly the same dosage level of the hormone and, of these, exactly 6 shows signs of aggression within one minute of receiving the dose.
(a) Calculate the probability of this happening for each of the three dosage levels, A; B and C: (This is essentially a Binomial random variable problem, so you can check your answers using EXCEL.) (b) Assuming that each of the three dosage levels was equally likely to have been administered in the …rst place (with a probability of 1=3), use Bayes’Theorem to evaluate the likelihood of each of the dosage levels given that 6 out of the 10 mice were observed to react in this way.
2. Let X be the random variable indicating the number of incoming planes every k minutes at a large international airport, with proba- bility mass function given by
p(x) = Pr(X = x) = (0:9k)x! xexp( 0:9k); x = 0; 1; 2; 3; 4; ::: . Find the probabilities that there will be
(a) exactly 9 incoming planes during a period of 5 minutes (i.e., …nd Pr(X = 9) when k = 5);
(b) fewer than 5 incoming planes during a period of 4 minutes (i.e., …nd Pr(X < 5) when k = 4);
(c) at least 4 incoming planes during an 2 minute period (i.e., …nd Pr(X 4) when k = 2).
5.7. EXERCISE 3 65
3. The random variable Y is said to be Geometric if it has probability mass function given by
p(y) = Pr(Y = y) = (1 ) y 1; y = 1; 2; 3; :::; 0 < < 1; where is an unknown ‘parameter’.
Show that the cumulative distribution function can be expressed as
P (y) = Pr(Y y) = 1 y; y = 1; 2; 3; ::: with P (y) = 0 for y 0 and P (y) ! 1 as y ! 1:
(Note that P (y) = p(1) + p(2) + ::: + p(y) =Pyt=1p(t) can be written in longhand as
P (y) = (1 ) 1 + + 2+ 3+ : : : + y 1 :
The term in the second bracket on the right-hand side is the sum of a Geometric Progression.)
4. The weekly consumption of fuel for a certain machine is modelled by means of a continuous random variable, X; with probability density function
g(x) = 3(1 x)
2; 0 x 1;
0; otherwise:
Consumption, X; is measured in hundreds of gallons per week.
(a) Verify thatR01g(x)dx = 1 and calculate Pr(X 0:5):
(b) How much fuel should be supplied each week if the machine is to run out fuel 10% of the time at most? (Note that if s denotes the supply of fuel, then the machine will run out if X > s:)
5. The lifetime of a electrical component is measured in 100s of hours by a random variable T having the following probability density function
f (t) = exp( t); t > 0; 0; otherwise:
(a) Show that the cumulative distribution function, F (t) = Pr(T t) is given by
F (t) = 1 exp( t); t > 0
0 t 0:
(b) Show the probability that a component will operate for at least 200 hours without failure is Pr(T 2) = 0:135:?
66CHAPTER 5. RANDOM VARIABLES & PROBABILITY DISTRIBUTIONS II
(c) Three of these electrical components operate independently of one another in a piece of equipment and the equipment fails if ANY ONE of the individual components fail. What is the probability that the equipment will operate for at least 200 hours without failure? (Use the result in (b) in a binomial context).
Chapter 6
THE NORMAL
DISTRIBUTION
It could be argued that the most important probability distribution en- countered thus far has been the Binomial distribution for a discrete random variable monitoring the total number of successes in n independent and iden- tical Bernoulli experiments. Indeed, this distribution was proposed as such by Jacob Bernoulli (1654-1705) in about 1700. However as n becomes large, the Binomial distribution becomes di¢ cult to work with and several math- ematicians sought approximations to it using various limiting arguments. Following this line of enquiry two other important probability distributions emerged; one was the Poisson distribution, due to the French mathematician Poisson (1781-1840), and published in 1837. An exercise using the Poisson distribution is provided by Question 2, in Exercise 3. The other, is the normal distribution due to De Moivre (French, 1667-1754), but more com- monly associated with the later German mathematician, Gauss (1777-1855), and French mathematician, Laplace (1749-1827). Physicists and engineers often refer to it as the Gaussian distribution. There a several pieces of evidence which suggest that the British mathematician/statistician, Karl Pearson (1857-1936) coined the phrase normal distribution.
Further statistical and mathematical investigation, since that time, has revealed that the normal distribution plays a unique role in the theory of statistics; it is without doubt the most important distribution. We introduce it here, and study its characteristics, but you will encounter it many more times in this, and other, statistical or econometric courses.
Brie‡y the motivation for wishing to study the normal distribution can be summarised in three main points:
it can provide a good approximation to the binomial distribution
it provides a natural representation for many continuous random vari- ables that arise in the social (and other) sciences
68 CHAPTER 6. THE NORMAL DISTRIBUTION
many functions of interest in statistics give random variables which have distributions closely approximated by the normal distribution.
We shall see shortly that the normal distribution is de…ned by a par- ticular probability density function; it is therefore appropriate (in the strict sense) for modelling continuous random variables. Not withstanding this, it is often the case that it provide an adequate approximation to another distribution, even if the original distribution is discrete in nature, as we shall now see in the case of a binomial random variable.
6.1
The Normal distribution as an approximation
to the Binomial distribution
Consider a Binomial distribution which assigns probabilities to the total number of successes in n identical applications of the same Bernoulli exper- iment. For the present purpose we shall use the example of ‡ipping a coin a number of times (n). Let the random variable of interest be the proportion of times that a HEAD appears and let us consider how this distribution changes as n increases:
If n = 3, the possible proportions could be 0; 1=3; 2=3 or 1
If n = 5, the possible proportions could be 0; 1=5; 2=5; 3=5; 4=5 or 1
If n = 10, the possible proportions could be 0; 1=10; 2=10; etc ...
The probability distributions, over such proportions, for n = 3; 5; 10 and 50; are depicted in Figure 7.1.
Notice that the ‘bars’, indicating where masses of probability are dropped, get closer and closer together until, in the limit, all the space between them is squeezed out and a bell shaped mass appears, by joining up the tops of every bar: this characterises the probability density function of the NOR- MAL DISTRIBUTION.
Having motivated the normal distribution via this limiting argument, let us now investigate the fundamental mathematical properties of this bell- shape.