distribution. The purpose of this section is to

gives the likelihood of simultaneously observing

introduce concepts that can be used to understand

a particular combination of hits and near-hits. The

the relationship between random variables in a

concepts introduced in Section 2.2 can be used to

multivariate setting. Marginal distributions [2.5.2]

show that this function is given by

± describe the properties of the individual random

C hn p h p n p (30’h’n)

30

variables that make up a random vector when the

HNM

fHN (h, n) = in¬‚uence of the other random variable in the ran-

for h + n ¤ 30 and h, n ≥ 0

dom vector is ignored. Conditional distributions

0 otherwise,

[2.5.4] describe the properties of some variable in

a random vector when variation in other parts of

where

the random variable is controlled.

C hn = 30!/ h!n!(30 ’ h ’ n)! ,

30

For example, we might be interested in

the distribution of rainfall when rainfall is

p H and p N are the probabilities of a hit and a near

forecast. If the forecast is skilful, this conditional

hit respectively, and

distribution will be different from the marginal

(i.e., climatological) distribution of rainfall. When

p M = (1 ’ p H ’ p N )

the forecast is not skilful (i.e., when the forecast

is the probability of a miss. is independent of what actually happens) marginal

2.5: Discrete Multivariate Distributions 27

X1 In example [2.5.1], the marginal distribution of

X1 is given in the row at the lower margin of

strong normal weak all

X2

Table 2.1, and that of X2 is given in the column

weak 21 11 2 34 at the right hand margin (hence the nomenclature).

moderate 20 14 7 41 The marginal distribution of X2 is

severe 4 4 6 14

±

very severe 0.34, x2 = weak

0 3 8 11

0.41, x2 = moderate

all 45 32 23 100 f X 2 (x2 ) =

0.14, x2 = strong

0.11, x2 = very strong.

Table 2.1: Estimated probability distribution (in

Note that f X 2 (weak), for example, is given by

%) of X = (X1 , X2 ) = (strength of westerly ¬‚ow,

severity of Baltic Sea ice conditions), obtained

f X 2 (weak) = f X (strong, weak)

from 104 years of data. Koslowski and Loewe

+ f X (normal, weak)

[231]. See [2.5.1].

+ f X (weak, weak)

= 0.21 + 0.11 + 0.02

and conditional distributions are identical. The

= 0.34.

effect of independence is described in [2.5.7].

2.5.1 Example. We will use the following

example in this section. Let X = (X1 , X2 )

2.5.4 Conditional Distributions. The concept

be a discrete bivariate random vector where X1

of conditional probability [2.2.5] is extended

takes values (strong, normal, weak) describing

to discrete random variables with the following

the strength of the winter mean westerly ¬‚ow in

de¬nition.

the Northeast Atlantic area, and X2 takes values

Let X1 and X2 be a pair of discrete random

(weak, moderate, severe, very severe) describing

variables. The conditional probability function of

the sea ice conditions in the western Baltic

X1 , given X2 = x2 , is

Sea (from Koslowski and Loewe [231]). The

probability distribution of the bivariate random

f X 1 X 2 (x1 , x2 )

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

variable is completely speci¬ed by Table 2.1. For (2.11)

f X 2 (x2 )

example: p(X1 = weak ¬‚ow and X2 = very severe

ice conditions) = 0.08. provided that f X 2 (x2 ) = 0.

Here f X 2 (x2 ) is the marginal distribution of X2

2.5.2 Marginal Probability Distributions. If which is given by f X 2 (x2 ) = f X 1 X 2 (x1 , x2 ).

X = (X1 , . . . , Xm ) is an m-variate random vector, The sum is taken over all possible realizations of

we might ask what the distribution of an individual (X1 , X2 ) for which X2 = x2 .

random variable Xi is if we ignore the presence

of the others. In the nomenclature of probability

2.5.5 Examples. The conditional distributions

and statistics, this is the marginal probability

for the example presented in Table 2.1 are derived

distribution. It is given by

by dividing row (or column) entries by the

f X i (xi ) = f (x1 . . . xi . . . xm ) corresponding row (or column) sum. For example,

the probability that the sea ice conditions are

x1 ,...,xi’1 ,xi+1 ,...,xm

severe given that the westerly ¬‚ow is strong is

where the sum is taken over all possible given by

realizations of X for which Xi = xi .

f (strong, severe)

f X 2 |X 1 =strong (severe) = X

f X 1 (strong)

2.5.3 Examples. If X has a multinomial

distribution, the marginal probability distribution 0.04

= = 0.09 .

of Xi is the binomial distribution with n trials and 0.45

probability pi of success. Consequently, if X ∼

In the rainfall forecast veri¬cation example

Mm (n, θ), with θ de¬ned as in [2.4.6], the mean

[2.4.5] the conditional distribution for the number

and variance of Xi are given by

of hits H given that there are N = m near hits is

µi = npi and σi = npi (1 ’ pi ). B(30 ’ m, p H /(1 ’ p N )).

2

2: Probability Theory

28

X1 marginal distribution for the strength of the

atmospheric ¬‚ow appears in the lowest row of

strong normal weak all

X2

Table 2.2. The changing climate is clearly re¬‚ected

weak 31 8 0 39 in the marginal distribution f˜X 2 , which is tabulated

moderate 30 10 4 44 in the right hand column. This suggests that weak

severe 6 3 3 12 and moderate ice conditions will be more frequent

very severe 0 2 4 6 in 2050 than at present, and that the frequency

all 67 23 11 101 of severe or very severe ice conditions will be

lowered from 25% to 18%.

Table 2.2: Hypothetical future distribution of X =

2.5.7 Independent Random Variables. The

(X1 , X2 ) = (strength of westerly ¬‚ow, severity of

idea of independence is easily extended to random

ice conditions), if the marginal distribution of the

variables because they describe events in the

westerly ¬‚ow is changed as indicated in the last

sample space upon which they are de¬ned. Two

row, assuming that no other factors control ice

random variables are said to be independent if they

conditions. (The marginal distributions do not sum

always describe independent events in a sample