Data Analysis and Statistics for Geography, Environmental Science, and Engineering

59

3

Random Variables,

Distributions, Moments,

and Statistics

3.1 RANDOM VARIABLES

We can dene a random variable (RV) once we have dened the sample space, based on the possible

events and their probabilities. An RV is a rule, or a function, or a map associating a number to each

event in the sample space (see Figure 3.1).

Example: In the roll of six-sided die, the events are A

i

= side facing up is i where i = 1, 2, …,

6 with P[A

i

] = 1/6. We can make a “discrete” RV, denoted by X, associating each event with

the value of the RV; for example, X taking values X = {1, 2, 3, 4, 5, 6} each with P[x

i

] = 1/6.

We refer to the type of RV described in the example as discrete because its values are discrete, that

is, a set of numbers, in this case integers 1, 2, …, 6. The events can be dened from intervals con-

tained in a range of real values from a to b. In this case, the values of RV X are continuous in this

range. We call this type of RV continuous.

Example: We measure concentration of a mineral (in ppm) at a given location and it can take

values between 0 and 10,000 ppm. Continuous random variable X is concentration. An event

could be dened as A = measured concentration is in the interval 10–15 ppm.

3.2 DISTRIBUTIONS

RV distributions are dened in the following manner. Here we assume that X is an RV.

3.2.1 probability Mass anD Density functions (pmf and pdf)

A discrete distribution or probability “mass” function (pmf) p(X) is a set of probabilities, one for

each value of X. More precisely, denoting x

i

as the values of X

px PX x

ii

()

[]==

(3.1)

for all values x

i

of X

01≤≤

px

i

() for all i (3.2)

px

i

i

()

∑

= 1 (3.3)

60 Data Analysis and Statistics for Geography, Environmental Science, and Engineering

The last equation says that the total probability (sample space) must be equal to 1 when all prob-

abilities are summed over all i.

Example: Toss a coin. Assign 0 to T and 1 to H. p(0) = 0.5, p(1) = 0.5. This is an example of a

uniform discrete RV: probabilities of each event are the same.

We can represent the probabilities as a graph as illustrated in Figure 3.2 for the example shown

earlier. Vertical thick arrows or bars represent a spike or impulse with intensity given by the height

of the spike and equals the probability of that particular value. Alternatively, it can be represented

as a bar graph where the height of each bar represents the probability (see Figure 3.2).

Example: Roll a six-side die. Then the pmf is p(x

i

) = 1/6 where x

i

= {1, 2, …, 6}. This is also a

uniform discrete RV (see Figure 3.3).

For more examples of illustrations of pmfs, see Davis, 2002, Chapter 2.

A continuous distribution or probability “density” function (pdf) p(X) is dened based on inter-

vals; the probability of the value being in an “innitesimal” (this is a calculus concept, basically

means “very, very small”) interval of X between x and x + dx, this to say

pxdx Px Xxdx() []=<≤+

(3.4)

123456

0.10

0.18

X

p(X)

1/6

123456

X

p(X)

0.00

0.10

0.20

1/6

FIGURE 3.3 pmf of a discrete RV for a six-side die. Spike and bar graphs.

0.0 0.2 0.4 0.6 0.8 1.0

0.3

0.5

0.7

X

p(X)

01

X

p(X)

0.0

0.2

0.4

FIGURE 3.2 pmf of a discrete RV represented as a spike graph and as bar graph.

U

B

A

X =0

X =1

Event spaceValues of random variables Probabilities

P(X =0)=

0.3

P(X =1)=

0.7

FIGURE 3.1 Constructing a random variable.