Subcriterio – Infraestructura

1. A and B are events such that Pr(A) = 0:4 and Pr(A [ B) = 0:75: (a) Find Pr(B) if A and B are mutually exclusive.

3.2. EXERCISE 2 39

2. Events A; B and C are such that B and C are mutually exclusive and Pr(A) = 2=3; Pr(A [B) = 5=6 and Pr(B [C) = 4=5: If Pr(BjA) = 1=2 and Pr(CjA) = 3=10; are A and C statistically independent?

3. Given the information in the example given in Section 3, about under- graduates taking Mathematics, Physics or Chemistry A-levels, calcu- late the following:

(a) Of those who took Mathematics, what proportion also took Physics (but not Chemistry) and what proportion took both Physics and Chemistry?

(b) Of those who took Physics and Chemistry, what proportion also took Mathematics?

4. The Survey of British Births, undertaken in the 1970s, aimed to im- prove the survival rate and care of British babies at, and soon after, birth by collecting and analysing data on new-born babies. A sam- ple was taken designed to be representative of the whole population of British births and consisted of all babies born alive (or dead) after the 24th week of gestation, between 0001 hours on Sunday 5 April and 2400 hours on Saturday 11 April 1970. The total number in the sample so obtained was n = 17; 530: A large amount of information was ob- tained, but one particular area of interest was the e¤ect of the smoking habits of the mothers on newly born babies. In particular, the ability of a newly born baby to survive is closely associated with its birth- weight and a birth-weight of less than 1500g is considered dangerously low. Some of the relevant data are summarised as follows.

For all new born babies in the sample, the proportion of mothers who: (i) smoked before and during pregnancy was 0:433

(ii) gave up smoking prior to pregnancy was 0:170 (iii) who had never smoked was 0:397:

However, by breaking down the sample into mothers who smoked, had given up, or who had never smoked, the following statistics were obtained:

(iv) 1:6% of the mothers who smoked gave birth to babies whose weight was less than 1500g,

(v) 0:9% of the mothers who had given up smoking prior to pregnancy gave birth to babies whose weight was less than 1500g,

(vi) 0:8% of mothers who had never smoked gave birth to babies whose weight was less than 1500g.

(a) Given this information, how would you estimate the risk, for a smoking mother, of giving birth to a dangerously under-weight

40 CHAPTER 3. CONDITIONAL PROBABILITY

baby? What is the corresponding risk for a mother who has never smoked? What is the overall risk of giving birth to an under-weight baby?

(b) Of the babies born under 1500g; estimate the proportion of these (a) born to mothers who smoked before and during pregnancy; (b) born to mothers who had never smoked.

(c) On the basis of the above information, how would you assess the evidence on smoking during pregnancy as a factor which could result in babies being born under weight?

5. Metal fatigue in an aeroplane’s wing can be caused by any one of three (relatively minor) defects, labelled A; B and C; occurring dur- ing the manufacturing process. The probabilities are estimated as: Pr(A) = 0:3; Pr(B) = 0:1; Pr(C) = 0:6: At the quality control stage of production, a test has been developed which is used to detect the presence of a defect. Let D be the event that the test detects a man- ufacturing defect with the following probabilities: Pr(DjA) = 0:6; Pr(DjB) = 0:2; Pr(DjC) = 0:7: If the test detects a defect, which of A; B or C is the most likely cause? (Hint : you need to …nd, and compare, Pr (AjD) ; Pr (BjD) and Pr (CjD) using Bayes Theorem.)

Chapter 4

RANDOM VARIABLES &

PROBABILITY

DISTRIBUTIONS I

The axioms of probability tell us how we should combine and use probabili- ties in order to make sensible statements concerning uncertain events. To a large extent this has assumed an initial allocation of probabilities to events of interest, from which probabilities concerning related events (unions, inter- sections and complements) can be computed. The question we shall begin to address in the next two sections is how we might construct models which assign probabilities in the …rst instance.

The ultimate goal is the development of tools which enable statistical analysis of data. Any data under consideration (after, perhaps, some cod- ing) are simply a set of numbers which describe the appropriate members of a sample in meaningful way. Therefore, in wishing to assign to probabilities to outcomes generated by sampling, we can equivalently think of how to as- sign probabilities to the numbers that are explicitly generated by the same sampling process. When dealing with numbers, a natural line of enquiry would be to characterise possible mathematical functions which, when ap- plied to appropriate numbers, yield probabilities satisfying the three basic axioms. A mathematical function which acts in this fashion is termed a mathematical or statistical model :

mathematical/statistical models: mathematical functions which may be useful in assigning probabilities in gainful way.

If such models are to have wide applicability, we need a ‘general’ ap- proach. As noted above, and previously, events (on a sample space of inter- est) are often described in terms of physical phenomena; see Sections 3 and 4. Mathematical functions require numbers. We therefore need some sort of mapping from the physical attributes of a sample space to real numbers,

42CHAPTER 4. RANDOM VARIABLES & PROBABILITY DISTRIBUTIONS I

before we can begin developing such models. The situation is depicted in Figure 5.1, in which events of interest de…ned on a physical sample space, S; are mapped into numbers, x; on the real line. Note that there is only one number for each physical event, but that two di¤erent events could be assigned the same number. Thus, this mapping can be described by a func- tion; it is this function, mapping from the sample space to the real line, which de…nes a random variable. A further function, f (:); is then applied on the real line in order to generate probabilities.

S

x

f(x)

Random Variable

tells us how to assign probabilities

Figure 4.1: Mapping from S to the real line

The initial task, then, is the mapping from S to the real line and this is supplied by introducing the notion of a random variable:

4.1

Random Variable

For our purposes, we can think of a random variable as having two compo- nents:

a label/description which de…nes the variable of interest

the de…nition of a procedure which assigns numerical values to events on the appropriate sample space.

Note that:

–often, but not always, how the numerical values are assigned will be implicitly de…ned by the chosen label

–A random variable is neither RANDOM or VARIABLE! Rather, it is device which describes how to assign numbers to physical events of interest: “a random variable is a real valued function de…ned on a sample space”.

4.1. RANDOM VARIABLE 43

–A random variable is indicated by an upper case letter (X, Y , Z, T etc). The strict mathematical implication is that since X is a function, when it is applied on a sample space (of physical attributes) it yields a number

The above is somewhat abstract, so let us now consider some examples of random variables:

4.1.1 Examples of random variables

Let X = ‘the number of HEADs obtained when a fair coin is ‡ipped 3 times’. This de…nition of X implies a function on the physical sam- ple space which generates particular numerical values. Thus X is a random variable and the values it can assume are:

X(H,H,H) = 3; X(T,H,H) = 2; X(H,T,H) = 2; X(H,H,T) = 2; X(H,T,T) = 1; X(T,H,T) = 1; X(T,T,H) = 1; X(T,T,T) = 0:

Let the random variable Y indicate whether or not a household has su¤ered some sort of property crime in the last 12 months, with Y (yes) = 1 and Y (no) = 0: Note that we could have chosen the numerical values of 1 or 2 for yes and no respectively. However, the mathematical treatment is simpli…ed if we adopt the binary responses of 1 and 0:

Let the random variable T describe the length of time, measured in weeks, that an unemployed job-seeker waits before securing permanent employment. So here, for example,

T (15 weeks unemployed ) = 5; T (31 weeks unemployed ) = 31; etc.

Once an experiment is carried out, and the random variable (X) is ap- plied to the outcome, a number is observed, or realised ; i.e., the value of the function at that point in the sample space. This is called a realisation, or possible outcome, of X and is denoted by a lower case letter, x:

In the above examples, the possible realisations of the random variable X (i.e., possible values of the function de…ned by X) are x = 0; 1; 2 or 3. For Y; the possible realisations are y = 0; 1; and for T they are t = 1; 2; 3; ::: .

The examples of X; Y and T given here all applications of discrete ran- dom variables (the outcomes, or values of the function, are all integers). Technically speaking, the functions X; Y and T are not continuous.

44CHAPTER 4. RANDOM VARIABLES & PROBABILITY DISTRIBUTIONS I