1

EngineeringComputation

Part 5

2

Some Concepts Previous to Probability

RANDOM EXPERIMENT

A random experiment or trial can be thought of as any activity that will result in one and only one of several well-defined outcomes, but one does not know in advance which one will occur.

SAMPLE SPACE

The set of all possible outcomes of a random experiment E, denoted by S(E), is called the sample space of the random experiment E.

EXAMPLE

Suppose that the structural condition of a concrete structure (e.g., a bridge) can be classified into one of three categories: poor, fair, or good. An engineer examines one such structure to assess its condition. This is a random experiment.

Its sample space, S(E) = {poor, fair, good}, has three elements.

If instead one measures the concrete quality in the range [0,10], this is the sample space.

3

Example of a random experiment

RANDOM EXPERIMENT

Rolling two dices

SAMPLE SPACE

The set {1,2,3,4,5,6} x {1,2,3,4,5,6}

4

Random Variable

RANDOM VARIABLE

A random variable can be defined as a real-valued function defined over a sample space of a random experiment. That is, the function assigns a real value to every element in the sample space of a random experiment.

The set of all possible values of a random variable X, denoted by S(X), is called the support or range of the random variable X.

EXAMPLE

In the previous concrete example, let X be −1, 0, or 1, depending on whether the structure is poor, fair, or good, respectively. Then X is a random variable with support S(X) = {−1, 0, 1}.

The condition of the structure can also be assessed using a continuous scale, say, from 0 to 10, to measure the concrete quality, with 0 indicating the worst possible condition and 10 indicating the best. Let Y be the assessed condition of the structure. Then Y is a random variable with support S(Y ) = {y : 0 ≤ y ≤ 10}

5

Random Variable

6

Random Variable

NOTATION

We consistently use the customary notation of denoting random variables by uppercase letters such as X, Y , and Z or X1,X2, . . . ,Xn, where n is the number of random variables under consideration. Realizations of random variables (that is, the actual values they may take) are denoted by the corresponding lowercase letters such as x, y, and z or x1, x2, . . . , xn.

DISCRETE AND CONTINUOUS RANDOM VARIABLES

A random variable is said to be discrete if it can assume only a finite or countably infinite number of distinct values. Otherwise, it is said to be continuous. Thus, a continuous random variable can take an uncountable set of real values.

The random variable X in the concrete example with possible values -1. 0, 1 is discrete whereas the random variable Y, with values in [0,10], is continuous.

UNIVARIATE AND MULTIVARIATE RANDOM VARIABLES

When we deal with a single random quantity, we have a univariate random variable.

When we deal with two or more random quantities simultaneously, we have a multivariate random variable.

7

Probability axioms

8

Probability properties

9

Induced probability of a random variable

10

Induced probability of a random variable

11

Induced probability of a random variable

12

Conditional probability

13

Independence of events

14

Total probability and Bayes theorems

15

Probability of a random variable

To specify a random variable we need to know:

1. its range or support, S(X), which is the set of all possible values of the random variable, and

2. a tool by which we can obtain the probability associated with every subset in its support, S(X). These tools are some functions such as the probability mass function (pmf), the cumulative distribution function (cdf)

16

Probability mass function of a discrete random variable

17

Cumulative distribution function of a discrete random variable

PropertiesConcrete example

18

Moments of a discrete random variable

19

Moments of a discrete random variable

20

Bernoulli Random Variable

21

Binomial random variable

The binomial random variable arises when one repeats n identical and independent Bernoulli experiments and observes the number of successes.

EXAMPLES

The number os cars taking left at one intersection of a series of 100 cars

The number of broken specimens in a test of a series of 100 specimens

The number of exceedances of a given flow level in a series of 365 days.

The number of waves higher than 10 m in a series of 1000 waves.

22

Binomial random variable

23

Binomial random variable

24

Binomial random variable

25

Binomial random variable

26

Geometric or Pascal random variable

27

Geometric or Pascal random variable

28

Return period

Thus, the return period is 1/p for exceedancesFor large values it becomes 1/(1-p)

29

Negative binomial random variable

30

Negative binomial random variable

31

Negative binomial random variable

32

Hypergeometric random variable

33

Hypergeometric random variable

34

Poisson random variable

Assumptions

35

Poisson random variable

36

Poisson random variable

37

Poisson random variable

38

Multivariate random variable

Joint probability mass function

Example

39

Marginal probability mass function

40

Conditional probability mass function

41

Variance and covariances

42

Means, variances and covariances

43

Covariance and correlation

44

Covariance and correlation

45

Multinomial distribution

46

Multinomial distribution