State (USA). we assume that our rainfall observations stem from

a stationary process, that is, that the likelihood

2.1.2 Simple Events and the Sample Space. of observing more, or less, than 0.1 inch daily

The sample space, denoted by S, is a list of rainfall is the same for all days within a winter and

possible outcomes of an experiment, where each the same for all winters. Observed records tell us

item in the list is a simple event, that is, an that the daily rainfall is greater than the 0.1 inch

experimental outcome that cannot be decomposed threshold on about 38 out of every 100 days. We

therefore estimate the relative likelihoods of the

into yet simpler outcomes.

For example, in the case of three consecutive two compound events in S.

tosses of a fair coin, the simple events are S As long as all outcomes are equally likely,

= {HHH, HHT, HTH, THH, TTH, THT, HTT, assigning probabilities can be done by counting

TTT} with H = ˜head™ and T = ˜tail.™ Another the number of outcomes in S. The sum of all

description of the possible outcomes of the coin the probabilities must be unity because one of the

tossing experiment is {˜three heads™, ˜two heads™, events in S must occur every time the experiment

˜one head™, ˜no heads™}. However, this is not a list is conducted. Therefore, if S contains M items, the

of simple events since some of the outcomes, such probability of any simple event is just 1/M. We see

as {˜two heads™}, can occur in several ways. below that this process of assigning probabilities

It is not possible, though, to list the simple by counting the number of elements in S can often

events that compose the West Glacier rainfall be extended to include simple events that do not

sample space. This is because a reasonable sample have the same likelihood of occurrence.

space for the atmosphere is the collection of all Once the probability of each simple event has

possible trajectories through its phase space, an been determined, it is easy to determine the

uncountably large collection of ˜events.™ Here we probability of a compound event. For example, the

19

2: Probability Theory

20

event {˜Heads on exactly 2 out of 3 tosses™} is partition it into subsets of simple events according

composed of the three simple events {HHT, HTH, to the number of successes. These compound

THH} and thus occurs with probability 3/8 on any events are made up of varying numbers of sample

repetition of the experiment. space elements. The smallest events (0 successes

and m successes) contain exactly one element

The word repetition is important because it

each. The next smallest events (one success in m

underscores the basic idea of a probability. If an

trials and m ’ 1 successes in m trials) contain

experiment is repeated ad in¬nitum, the proportion

m elements each. In general, the event with n

of the realizations resulting in a particular outcome

successes in m trials contains

is the probability of that outcome.

m!

m=

n n!(m ’ n)!

2.2 Probability

simple events. These compound events do not

2.2.1 Discrete Sample Space. A discrete

contain any common elements, so it follows that

sample space consists of an enumerable collection

m

m

n=1 n = 2 .

m

of simple events. It can contain either a ¬nite or a

countably in¬nite number of elements.

An example of a large ¬nite sample space occurs

2.2.3 A Sample Space is More Than a

when a series of univariate statistical tests (see

Collection of Simple Events. A complete

[6.8.1]) is used to validate a GCM. The test makes

probabilistic description of an experiment must be

a decision about whether or not the simulated

more than just a list of simple events. We also

climate is similar to the observed climate in each

need a rule, say P (·), that assigns probabilities

model grid box (Chervin and Schneider [84];

to events. In simple situations, such as the coin

Livezey and Chen [257]; Zwiers and Boer [446]).

tossing example of Section 2.1, P (·) can be based

If there are m grid boxes (m is usually of order 103

on the numbers of elements in an event.

or larger), then the number of possible outcomes

Different experiments may generate the same

of the decision making procedure is 2m ”a large

set of possible outcomes but have different rules

but ¬nite number. We could be exhaustive and list

for assigning probabilities to events. For example,

each of the 2m possible ¬elds of decisions, but it is

a fair and a biased coin, each tossed three times,

easy and convenient to characterize more complex

generate the same list of possible outcomes but

events by means of a numerical description and to

each outcome does not occur with the same

count the number of ways each can occur.1

likelihood. We can use the same threshold for

An example of an in¬nite discrete sample

daily rainfall at every station and will ¬nd different

space occurs in the description of a precipitation

climatology, where S = {0, 1, 2, 3, . . .} lists the likelihoods for the exceedance of that threshold.

waiting times between rain days.2

2.2.4 Probability of an Event. The probability

2.2.2 Binomial Experiments. Experiments of an event in a discrete sample space is computed

analogous to the coin tossing, rainfall threshold by summing up the probabilities of the individual

exceedance, and testing problems described above sample space elements that comprise the event.

are particularly important. They are referred to as A list of the complete sample space is usually

binomial experiments because each replication of unnecessary. However, we do need to be able to

the experiment consists of a number of Bernoulli enumerate events, that is, count elements in subsets

of S.

trials; that is, trials with only two possible

outcomes (which can be coded ˜S™ and ˜F™ for Some basic rules for probabilities are as follows.

success and failure).

• Probabilities are non-negative.

An experiment that consists of m Bernoulli trials

has a corresponding sample space that contains 2m

• When an experiment is conducted, one of the

entries. One way to describe S conveniently is to

simple events in S must occur, so

1 We have taken some liberties with the idea of a discrete

P (S) = 1.

sample space in this example. In reality, each of the ˜simple

events™ in the sample space is a compound event in a very large

(but discrete) space of GCM trajectories.

• It may be easier to compute the probability

2 We have taken additional liberties in this example. The

of the complement of an event than that of

events are really compound events in the uncountably large

the event itself. If A denotes an event, then

space of trajectories of the real atmosphere.

2.3: Discrete Random Variables 21

¬A, its complement, is the collection of all 2.2.6 Independence. Two events A and B are

elements in S that are not contained in A. said to be independent of each other if

That is, S = A ∪ ¬A. Also, A © ¬A = ….

P (A © B) = P (A)P (B). (2.3)

Therefore,

It follows from (2.2) that if A and B are

P (A) = 1 ’ P (¬A).

independent, then P (A|B) = P (A). That is,

restriction of the sample space to B gives no

• It is often useful to divide an event into additional information about whether or not A will

smaller, mutually exclusive events. Two occur.

events A and B are mutually exclusive if they Suppose A represents severe weather and B

do not contain any common sample space represents a 24-hour forecast of severe weather.

elements, that is, if A© B = …. An experiment If A and B are independent, then the forecasting

can not produce two mutually exclusive system does not produce skilful severe weather

outcomes at the same time. Therefore, if A forecasts: a severe weather forecast does not

and B are mutually exclusive, change our perception of the likelihood of severe

weather tomorrow.

P (A ∪ B) = P (A) + P (B). (2.1)

2.3 Discrete Random Variables

• In general, the expression for the probability

of observing one of two events A and B is 2.3.1 Random Variables. We are usually not

really interested in the sample space S itself, but

P (A ∪ B) = P (A) + P (B) ’ P (A © B). rather in the events in S that are characterized by

functions de¬ned on S. For the three coin tosses