Probability Distributions - Duke University

Probability Distributions

CEE 201L. Uncertainty, Design, and Optimization Department of Civil and Environmental Engineering Duke University

Philip Scott Harvey, Henri P. Gavin and Jeffrey T. Scruggs Spring 2024

In the context of random variables, capital italics (X) represent an uncertain quantity (a random variable) and lower case italics (x) represent a particular value of that random variable. Random variables can be discrete or continuous.

? Discrete random variables can take on values that are members of a (finite or infinite) set of discrete values. If X can take on only positive whole numbers (the number of times a team can win over all time), then X is a discrete random variable with an infinitely large population. If X can take on only whole numbers, between 0 and 23, (the hour of a day) then X is a discrete random variable with a finite population.

? Continuous random variables can take on any value within finite or infinite bounds. The population of potential values of any continuous random variable is infinitely large.

This document focuses on continuous random variables.

1 Probability distributions of continuous random variables

The properties of a random variable (rv) X distributed over the domain x X x^ are fully described by its probability density function or its cumulative distribution function.

The probability density function (PDF) of X is the function fX (x) such that for any two numbers a and b within the domain x a b x^,

b

P [a < X b] = fX (x) dx

a

For fX (x) to be a proper distribution, it must satisfy the following two conditions:

? The PDF fX (x) is not negative; fX (x) 0 for all values of x between x and x^.

x^

? The rule of total probability holds; the total area under fX (x) is 1; fX (x) dx = 1.

x

The cumulative distribution function (CDF) of X is the function FX (x) that gives, for any specified value b between x and x^, the probability that the random variable X is less than or equal to the value b is written as P [X b]. The CDF is defined by

x

FX (x) = P [X x] = fX (s) ds ,

-

2 CEE 201L. Uncertainty, Design, and Optimization ? Duke University ? Spring 2024 ? P.S.H., H.P.G. and J.T.S.

where s is a dummy variable of integration. So, P [a < X b] = FX (b) - FX (a)

By the first fundamental theorem of calculus, the functions fX (x) and FX (x) are related as d

fX (x) = dx FX (x) Some important characteristics of CDF's and PDF's of X are:

? CDF's, FX (x), are monotonic non-decreasing functions of x.

? For any number a, P [X > a] = 1 - P [X a] = 1 - FX (a)

b

? For any two numbers a and b, with a b, P [a < X b] = FX (b) - FX (a) = fX (x)dx

a

? The likelihood L of an event (e.g., an observation of X that reveals a value of x) is defined as the value of the probability density of X at the observed value of x.

L(x) fX (x) .

Seen another way, the likelihood could be the probability that the observation lies between x and x + dx, in the limit that dx goes to zero

x+dx

L(x) = lim

dx0 x

fX (x) dx fX (x) dx .

and usually the dx is simply dropped for convenience.

PDF : fX(x)

0.35 0.3

0.25 0.2

0.15 0.1

0.05 0

0

P[X ................
................

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