1 Engineering Computation 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 NOTATION We consistently use the customary notation of denoting random variables by uppercase letters such as X, Y, and Z or X 1,X 2,...,X n, 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 x 1, x 2,..., x n. 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 Properties Concrete 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 exceedances For 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