Distribution and Density Functions

A.5.2 Distribution and Density Functions

The probability distribution function (PDF) of a random variable X is defined as:

F X ( x ) = P ( X ≤ x ) . A. 17

We usually simplify the notation, whenever no confusion can arise from its use,

by writing F (x ) instead of F X (x ) .

Figure A.2 shows the distribution function of the random variable X(a, b) =

a + b of Example A.10. Until now we have only considered examples involving sample spaces with a finite number of elementary events, the so-called discrete sample spaces to which discrete random variables are associated. These can also represent a denumerably infinite number of elementary events.

For discrete random variables, with probabilities p j assigned to the singleton events of A, the following holds:

F ( x ) = ∑ p j . A. 18

For instance, in Example A.10, we have F(4.5) = p 1 + p 2 + p 3 = 0.17 with p 1 = P({(1,1)}, p 2 = P({(1,2), (2,1)}) and p 3 = P({(1,3), (2,2), (3,1)}). The p j

sequence is called a probability distribution. When dealing with non-denumerable infinite sample spaces, one needs to resort to continuous random variables, characterized by a continuous distribution function F X (x), differentiable everywhere (except perhaps at a finite number of points).

412 Appendix A - Short Survey on Probability Theory

Figure A.2. Distribution function of the random variable associated to the sum of the faces in the two-dice throwing experiment. The solid circles represent point inclusion.

The function f X ( x ) = dF X ( x ) / dx (or simply f(x)) is called the probability

density function (pdf) of the continuous random variable X. The properties of the density function are as follows:

i.

f ( x ) ≥ 0 ( where defined ) ;

ii. ∫ − ∞ f ( dt t ) =1 ;

iii. x F ( x ) = ∫

− ∞ f ( t ) dt .

The event corresponding to ] a, b] has the following probability:

a f ( t ) dt . A. 19

This is the same as P ( a ≤ X ≤ b ) in the absence of a discontinuity at a. For an

infinitesimal interval we have:

F ( a + ∆ a ) − F ( a ) P ( [ a , a + ∆ a ] ) A. 20

which justifies the name density function, since it represents the “mass” probability corresponding to the interval ∆a, measured at a, per “unit length” of the random variable (see Figure A.3a).

The solution X = x α of the equation:

F X (x ) = α , A. 21 is called the α-quantile of the random variable X. For α = 0.1 and 0.01, the

quantiles are called deciles and percentiles. Especially important are also the quartiles ( α = 0.25) and the median (α = 0.5) as shown in Figure A.3b. Quantiles

are useful location measures; for instance, the inter-quartile range, x 0.75 –x 0.25 , is

often used to locate the central tendency of a distribution.

A.5 Random Variables and Distributions 413

Figure A.3.

a) A pdf example; the shaded area in [a, a+ ∆a] is an infinitesimal probability mass. b) Interesting points of a pdf: lower quartile (25% of the total area); median (50% of the total area); upper quartile (75% of the total area).

a -0.2 0 0.2 0.4 0.6 0.8 1 1.2 b -0.2 0 0.2 0.4 0.6 0.8 1 1.2

Figure A.4. Uniform random variable: a) Density function (the circles indicate point inclusion); b) Distribution function.

Figure A.4 shows the uniform density and distribution functions defined in

[0, 1]. Note that P ( a < X ≤ a + w ) = w for every a such that [ a, a +w ] ⊂ [0, 1],

which justifies the name uniform distribution.