space. More precisely:

[2.5.6].

Two random variables, X1 and X2 , are said to be

˜independent™ if

2.5.6 Example: Climate Change and Western

f X 1 ,X 2 (x1 , x2 ) = f X 1 (x1 ) f X 2 (x2 )

Baltic Sea-ice Conditions. In [2.5.5] we sup- (2.12)

posed that sea-ice conditions depend on atmo-

for all (x1 , x2 ).

spheric ¬‚ow. Here we assume that atmospheric

That is, two random variables are independent if

¬‚ow controls the sea-ice conditions and that feed-

their joint probability function can be written as

back from the sea-ice conditions in the Baltic Sea,

the product of their marginal probability functions.

which have small scales relative to that of the

Using (2.11) and (2.12) we see that indepen-

atmospheric ¬‚ow, may be neglected. Then we can

dence of X1 and X2 implies

view the severity of the ice conditions, X2 , as being

dependent on the atmospheric ¬‚ow, X1 .

f X 1 |X 2 =x2 (x1 ) = f X 1 (x1 ).

Table 2.1 seems to suggest that if stronger

westerly ¬‚ows were to occur in a future climate, Thus, knowledge of the value of X2 does not

give us any information about the value of X1 .6

we might expect relatively more frequent moderate

and weak sea-ice conditions. The next few A useful result of (2.12) is that, if X1 and X2 are

subsections examine this possibility. independent random variables, then

We represent present day probabilities with

E(X1 X2 ) = E(X1 )E(X2 ). (2.13)

the symbol f and those of a future climate,

in say 2050, by f˜. We assume that conditional The reverse is not true: nothing can be said about

probabilities are unchanged in the future, that is, the independence of X1 and X2 when (2.13) holds.

f X 2 |X 1 =x1 (x2 ) = f˜X 2 |X 1 =x1 (x2 ). However, if (2.13) does not hold, X1 and X2 are

certainly dependent.

Using (2.11) to express the joint present and

future probabilities as products of the conditional

2.5.8 Examples. The two variables described

and marginal distributions, we ¬nd

in Table 2.1 are not independent of each other

f˜X 1 (x1 ) because the table entries are not equal to the

f˜X (x1 , x2 ) = f X (x1 , x2 ).

f X 1 (x1 ) product of the marginal entries. Thus, knowledge

of the value of the westerly ¬‚ow index, X1 , tells

Now suppose that the future marginal probabilities

you something useful about the relative likelihood

for the atmospheric ¬‚ow are f˜X 1 (strong) =

that the different values of sea-ice intensity X2 will

0.67, f˜X 1 (normal) = 0.22 and f˜X 1 (weak) = be observed.

0.11. Then the future version of Table 2.1 What would Table 2.1 look like if the strength

is Table 2.2.5 Note that the prescribed future of the westerly ¬‚ow, X , and the severity of

1

the Western Baltic sea-ice conditions, X2 , were

5 These numbers were derived from a ˜doubled CO

2

experiment™ [96]. Factors other than atmospheric circulation independent? The answer, assuming that there is

probably affect the sea ice signi¬cantly, so this example should

6 Thus the present de¬nition is consistent with [2.2.6].

not be taken seriously.

2.6: Continuous Random Variables 29

X1 Thus, the mean of the sum of n independent

identically distributed random variables is n times

strong normal weak all

X2

the mean of the individual random variable.

weak 15 11 8 34 Likewise, the variance of the sum is n times the

moderate 18 13 9 40 variance of X.

severe 6 4 3 13

very severe 5 4 4 13

2.6 Continuous Random Variables

all 44 32 24 100

2.6.0 Introduction. Up to this point we have

discussed examples in which, at least conceptually,

Table 2.3: Distribution of X = (X1 , X2 ) =

we can write down all the simple outcomes of an

(strength of westerly ¬‚ow, severity of ice condi-

experiment, as in the coin tossing experiment or

tions) assuming that the severity of the sea-ice

in Table 2.1. However, usually the sample space

conditions and the strength of the westerly ¬‚ow

cannot be enumerated; temperature, for example,

are unrelated. See [2.5.8]. (Marginal distribution

varies continuously.7

deviates from that of Table 2.1 because of rounding

errors.)

2.6.1 The Climate System™s Phase Space. We

have discussed temperature measurements in the

no change in the marginal distributions, is given in context of a sample space to illustrate the idea of a

continuous sample space”but the idea that these

Table 2.3.

The two variables described by the bivariate measurements de¬ne the sample space, no matter

multinomial distribution [2.4.5] are also depen- how ¬ne the resolution, is fundamentally incorrect.

dent. One way to show this is to demonstrate Temperature (and all other physical parameters

that the product of the marginal distributions is used to describe the state of the climate system)

not equal to the joint distribution. Another way should really be thought of as functions de¬ned on

to show this is to note that the set of values that the climate™s phase space.

The exact characteristics of phase space are not

can be taken by the random variable pair (H, N)

is not equivalent to the cross-product of the sets of known. However, we assume that the points in the

values that can be taken by H and N individually. phase space that can be visited by the climate are

For example, it is possible to observe H = n not enumerable, and that all transitions from one

or N = n separately, but one cannot observe part of phase space to another occur smoothly.

The path our climate is taking through phase

(H, N) = (n, n) because this violates the condition

space is conceptually one of innumerable paths.

that 0 ¤ H + N ¤ n.

If we had the ability to reverse time, a small

change, such as a slightly different concentration

2.5.9 Sum of Identically Distributed Inde- of tropospheric aerosols, would have sent us down

pendent Random Variables. If X is a random a different path through phase space. Thus, it is

variable from which n independent realizations xi perfectly valid to consider our climate a realization

n

are drawn, then y = i=1 xi is a realization of the of a continuous stochastic process even though the

n

random variable Y = i=1 Xi , where the Xi s are time-evolution of any particular path is governed

independent random variables, each distributed as by physical laws. In order to apply this fact to our

X. Using independence, it is easily shown that the diagnostics of the observed and simulated climate

we have to assume that the climate is ergodic.

mean and the variance of Y are given by

That is, we have to assume that every trajectory

will eventually visit all parts of phase space and

E(Y) = n E(X)

that sampling in time is equivalent to sampling

Var(Y) = E Y2 ’ E(Y)2

different paths through phase space. Without this

n

assumption about the operation of our physical

= E Xi X j ’ n 2 E(X)2

system the study of the climate would be all but

i, j=1

impossible.

= n E X2 + n(n ’ 1)E(X)2

7 In reality, both the instrument used to take the

’ n E(X)

2 2

measurement and the digital computing system used to store

= n E X2 ’ E(X)2 it operate at ¬nite resolutions. However, it is mathematically

convenient to approximate the observed discrete random

= n Var(X). variable with a continuous random variable.

2: Probability Theory

30

The conclusion is that our initial assumption, that

The assumption of ergodicity is well founded,

there is a point x for which P (X = x) > 0, is false.

at least on shorter time scales, in the atmosphere

That is, if X is a continuous random variable, then

and the ocean. In both media, the laws of physics