Probability Models

Abstract

This chapter provides a brief introduction to some important probability models. It starts from the basic concept of probability space and random variables, cumulative distribution function, probability density function/probability mass function, moment generating function and characteristic function, followed by selected frequently-used distribution types with illustration of their applications in practical engineering. The joint probabilistic behaviour of different random variables is also discussed, including the use of copula function to construct the joint distribution functions of dependent random variables.

Problems

2.1

Consider the damage state of a post-hazard structure, which be classified into five categories: none ($D_0$), minor ($D_1$), moderate ($D_2$), severe ($D_3$) and total ($D_4$). The post-hazard structural performance is deemed as “unsatisfactory” if the damage reaches a moderate or severer state. Let $E_1$ be the event that the structure suffers from damage, and $E_2$ the event that the structural performance is unsatisfactory.

(1) What is the sample space? What are the event spaces for $E_1$ and $E_2$ respectively?

(2) If the probability of $D_i$ ($i=0,1,2,3,4$) is proportional to $(i+1)^{-1}$, compute the probabilities of $E_1$ and $E_2$ respectively.

2.2

Consider two post-hazard structures, namely 1 and 2, whose performances are as described in Problem 2.1. Let $p_{1}$ be the probability that the structural performance is unsatisfactory. Furthermore, let $\widetilde{E}_1$ be the event that both structural performances are unsatisfactory, and $\widetilde{E}_2$ the event that either structural performance is unsatisfactory.

(1) What is the sample space for the joint behaviour of the two post-hazard structures? What are the event spaces for $\widetilde{E}_1$ and $\widetilde{E}_2$?

(2) Derive the lower bound for $\mathbb {P}(\widetilde{E}_1)$ in terms of $p_{1}$.

(3) Derive the upper bound for $\mathbb {P}(\widetilde{E}_2)$ in terms of $p_{1}$.

(4) If the structural performances are independent, and the probability of $D_i$ ($i=0,1,2,3,4$) is proportional to $(i+1)^{-1}$, compute the probabilities of $\widetilde{E}_1$ and $\widetilde{E}_2$ respectively, and verify the results in (2) and (3).

2.3

Consider a structure subjected to repeated loads. In the presence of the effect of each load, the structure may fail in either mode 1 or mode 2. Let $p_1$ and $p_2$ denote the probability of failure associated with mode 1 and mode 2 respectively due to the effect of one load. Show that, if the structural fails, the probability of mode 1 failure is $\frac{p_1}{p_1+p_2}$.

2.4

In Problem 2.1, if the probability of $D_i$ ($i=0,1,2,3,4$) is $p_i$, what is the probability of severe damage conditional on the fact that the structural performance is unsatisfactory? What is the conditional probability of $E_2$ on $E_1$?

2.5

Recall Example 2.7.

(1) Determine the probability that V occurs only conditional on the failure mode of “beam”.

(2) Determine the probability that V occurs only conditional on the failure of the frame (in any mode).

(3) Compare the two probabilities obtained in (1) and (2).

2.6

Consider the resistance (moment-bearing capacity) of a bridge girder, which follows a lognormal distribution with a mean value of 4000 kN$\cdot$m and a standard deviation of 400 kN$\cdot$m.

(1) If the bridge girder has survived a load of 3700 kN$\cdot$m, compute the mean value and standard deviation of the post-load resistance.

(2) If the bridge girder has survived a load with a mean value of 3700 kN$\cdot$m and a standard deviation of 370 kN$\cdot$m, compute the mean value and standard deviation of the post-load resistance.

(3) Compare the results in (1) and (2), and comment on the difference.

2.7

Recall Example 2.8. If the bridge girder have survived two independent proof loads, namely $S_1$ and $S_2$, with CDFs of $F_{S_1}(s)$ and $F_{S_2}(s)$ respectively, determine the updated distribution of girder resistance.

2.8

Consider two post-hazard structures, namely 1 and 2, whose performances are as described in Problem 2.1. Let $E_i^*$ be the event that the performance of structure i is satisfactory for $i=1,2$. Show that $\mathbb {P}(E_1^*|E_2^*)=\mathbb {P}(E_2^*|E_1^*)$.

2.9

In Problem 2.1, suppose that the probability of $D_i$ is proportional to $(i+1)^{-1}$, and that the economic loss of the structure is $L_i$ conditional on $D_i$ for $i=0,1,2,3,4$. Assume that $L_i=ki^2$ for $i=0,1,2,3,4$, where $k>0$ is a constant. We introduce a random variable X to represent the post-hazard damage state, and a random variable Y to represent the post-hazard economic loss. The variable X takes a value of i corresponding to the damage state $D_i$.

(1) What are the probability mass function, mean and standard deviation of X?

(2) Compute $\mathbb {P}(1<X\le 3)$.

(3) What are the mean and standard deviation of Y?

2.10

Consider the post-hazard damage state of a building portfolio with 100 buildings. Each building has an identical failure probability of p, and the behaviour of each building is assumed to be independent of others. The probability that at least 95% of the post-hazard buildings remain safe is required to be greater than 99%. What is the range for p?

2.11

Suppose that random variable X has a mean value of 3 and its PDF is

$$\begin{aligned} f_X(x)=a\exp \left( -(x-b)^2\right) ,\quad -\infty<x<\infty \end{aligned}$$

(2.371)

(1) Compute a and b;

(2) What is the standard deviation of X?

(3) Compute the mean value of $\exp (X)$.

2.12

For two variables $X,Y\in [0,1]$, if their joint PDF is $f_{X,Y}(x,y)=a xy^2\cdot \mathbb {I}(0\le x\le 1, 0\le y\le 1)$, where a is a constant, decide whether X and Y are independent or not.

2.13

Markov’s inequality and Chebyshev’s inequality

(1) Suppose that the tensile strength of a steel bar is a random variable with a mean value of 300 MPa. Using the Markov’s inequality, what is the upper bound for the event that the tensile strength exceeds 600 MPa?

(2) Suppose that the tensile strength of a steel bar is a random variable with a mean value of 300 MPa and a standard deviation of 50 MPa. Using the Chebyshev’s inequality, what is the lower bound for the event that the tensile strength is within [250 Mpa, 350 Mpa]?

2.14

Consider the post-hazard damage state of a building portfolio with 100 buildings. Each building has independent performance and an identical probability of 0.1 for unsatisfactory performance. Let X be the number of buildings with unsatisfactory performance. Compute the probability of $X\le 40$.

2.15

Reconsider Problem 2.14.

(1) Compute the probability of $X=40$.

(2) Using a normal distribution to approximate X, recalculate the probability of $X=40$ [Hint. The probability can be approximately replaced by $\mathbb {P}(39.5\le X<40.5)$.].

(3) Compare the results in (1) and (2).

2.16

Suppose that random variable X is binomially distributed with parameters n and p. For $k=0,1,\ldots , n$, what is the maximum value for $\mathbb {P}(X = k)$?

2.17

Repeating the experiment in Sect. 2.2.1.1 until the rth outcome of success, the number of experiments, denoted by X, follows a negative binomial distribution, and its PMF is

$$\begin{aligned} \mathbb {P}(X=k)=\mathsf {C}_{k-1}^{r-1}p^r(1-p)^{k-r},\quad k\ge r \end{aligned}$$

(2.372)

(1) What are the mean value and variance of X?

(2) Derive the MGF of X.

2.18

Suppose that a Poisson random variable X has a mean value of $\lambda$. For $k=0,1,2,\ldots$, what is the maximum value for $\mathbb {P}(X = k)$?

2.19

For two independent binomial random variables X (with parameters $p,n_1$) and Y (with parameters $p,n_2$), show that $X+Y$ is also a binomial random variable.

2.20

For two independent random variables X and Y with mean values of $\mu _X,\mu _Y$ and standard deviations of $\sigma _X,\sigma _Y$, derive $\mathbb {V}(XY)$ in terms of $\mu _X,\mu _Y,\sigma _X$ and $\sigma _Y$.

2.21

If random variable X is uniformly distributed within [0, 1],

(1) Show that $X^n$ follows a Beta distribution, where n is a positive integer.

(2) Compute $\mathbb {E}(X^n)$ and $\mathbb {V}(X^n)$.

2.22

For two random variables X and Y that are uniformly distributed within [0, 1], find the PDF of $X+Y$ [Hint. $X+Y$ follows a triangle distribution.].

2.23

Let N be a Poisson random variable with a mean value of $\lambda$, and $X_1,X_2,\ldots , X_N$ a sequence of statistically independent and identically distributed random variables (independent of N). Let $Y=\sum _{i=1}^{N} X_i$, then Y follows a compound Poisson distribution. If the MGF of each $X_i$ is $\psi _X(\tau )$, derive the MGF of Y.

2.24

For a random variable $X\in (0,1)$ whose PDF is as follows, it is deemed to follow a standard arcsine distribution,

$$\begin{aligned} f_X(x)=\frac{1}{\pi \sqrt{x(1-x)}} \end{aligned}$$

(2.373)

Find the mean value and standard deviation of X respectively [Hint. This corresponds to a special case of $\eta =\zeta =\frac{1}{2}$ in Eq. (2.223).].

2.25

A random variable X follows a Laplace distribution with parameters $\mu$ and $b>0$ if its PDF is as follows,

$$\begin{aligned} f_X(x)=\frac{1}{2b}\exp \left( -\frac{|x-\mu |}{b}\right) ,\quad -\infty<x<\infty \end{aligned}$$

(2.374)

Show that (1) the mean of X is b; (2) the variance of X is $2b^2$; (3) the MGF of X is $\frac{\exp (\mu \tau )}{1-b^2\tau ^2}$ for $|\tau |< \frac{1}{b}$.

2.26

An Erlang distributed random variable X with shape parameter $k\in \{1,2,3,\ldots \}$ and rate parameter $r\ge 0$ has the following PDF,

$$\begin{aligned} f_X(x)=\frac{r^k x^{k-1}\exp (-rx)}{(k-1)!}, \quad x\ge 0 \end{aligned}$$

(2.375)

(1) What are the mean value and standard deviation of X?

(2) What is the MGF of X?

(3) Show that for two Erlang variables $X_1$ and $X_2$ with the same rate parameter, $X_1+X_2$ is also an Erlang random variable [Remark. The Erlang distribution is a special case of the Gamma distribution with the shape parameter being an integer.].

2.27

Let $X_1,X_2,\ldots , X_n$ ($n\ge 2$) be a sequence of statistically independent and identically distributed continuous random variables with a CDF of $F_X(x)$ and a PDF of $f_X(x)$. Let Y be the largest variable among $\{X_i,i=1,2,\ldots , n\}$, and W the smallest variable among $\{X_i\}$.

(1) What are the CDF and PDF of Y?

(2) Use the result from (1) to resolve Problem 2.7.

(3) What are the CDF and PDF of W?

2.28

Suppose that $X_1,X_2,\ldots , X_n$ is a sequence of independent geometrically distributed variables. Show that $Y=\min _{i=1}^n X_i$ also follows a geometric distribution.

2.29

For a discrete random variable X that is geometrically distributed with parameter p, show that as $p\rightarrow 0$, the distribution of pX approaches an exponential distribution.

2.30

Reconsider Problem 2.27. Let Z be the second-largest variable among $\{X_i,i=1,2,\ldots , n\}$. What are the CDF and PDF of Z?

2.31

Recall Eq. (2.162), which gives the CDF of the arriving time of the first event, $T_1$. Now, let $T_2$ be the arriving time of the second event. What are the PDF, mean and variance of $T_2$?

2.32

The water seepage in a lining structure can be modeled by the Darcy’s law as follows,

$$\begin{aligned} \chi (t)=\sqrt{2K\frac{p}{\omega _0}t} \end{aligned}$$

(2.376)

where $\chi (t)$ is the depth of water seepage at time t, K is the hydraulic conductivity, p is the water pressure on the top of the lining structure, and $\omega _0$ is the water density [34]. Treating p and $\omega _0$ as constants, show that if K is lognormally distributed, then $\chi (t)$ also follows a lognormal distribution for $t>0$.

2.33

Show that if X is an exponentially distributed variable, then $Y=\sqrt{X}$ follows a Rayleigh distribution.

2.34

Let $X_1,X_2,\ldots , X_n$ be a sequence of statistically independent and identically distributed variables. If each $X_i$ follows a Rayleigh distribution with a scale parameter of $\sigma$, show that $Y=\sum _{i=1}^{n}X_i^2$ is a Gamma random variable.

2.35

If X is an exponentially distributed variable, show that $-\ln X$ follows a Gumbel distribution.

2.36

The CDF of the generalized extreme value (GEV) distribution is as follows,

$$\begin{aligned} F(x)=\exp \left( -(1+\xi x)^{-\frac{1}{\xi }}\right) ,\quad 1+\xi x>0 \end{aligned}$$

(2.377)

For the cases of $\xi <0$, $\xi \rightarrow 0$ and $\xi >0$ respectively, what distribution does Eq. (2.377) reduce to?

2.37

For a Pareto (Type I) random variable X with parameters $\alpha >0$ and $x_m>0$, its CDF takes a form of

$$\begin{aligned} F_X(x)=\left\{ \begin{aligned}&1-\left( \frac{x}{x_m}\right) ^\alpha ,&x\ge x_m \\&0,&x<x_m\end{aligned}\right. \end{aligned}$$

(2.378)

(1) Derive the mean value and variance of X.

(2) Show that $\ln \left( \frac{X}{x_m}\right)$ is an exponential random variable.

2.38

The CDF of the generalized Pareto distribution (GPD) is defined as follows,

$$\begin{aligned} F(x)=1-(1+\xi x)^{-\frac{1}{\xi }},\quad 1+\xi x>0 \end{aligned}$$

(2.379)

Derive the mean value and variance of the GPD in Eq. (2.379) [Remark. The GPD is often used to describe a variable’s tail behaviour, and is an asymptotical distribution of the GEV distribution as in Eq. (2.377).].

2.39

A random variable X is said to follow a standard Cauchy distribution if its PDF is

$$\begin{aligned} f_X(x)=\frac{1}{\pi (1+x^2)} \end{aligned}$$

(2.380)

Show that for two independent standard normal random variables $Y_1$ and $Y_2$ (with a mean value of 0 and a variance of 1), $\frac{Y_1}{Y_2}$ follows a standard Cauchy distribution [Hint. Consider the joint distribution of $U=\frac{Y_1}{Y_2}$ and $V=Y_2$ using the Jacobian matrix.].

2.40

Reconsider Problem 2.23. Compute $\mathbb {C}(N,Y)$.

2.41

Suppose that random variable $X\in [0,1]$ has a mixture distribution with $\mathbb {P}(X=0)=p_0$ and $\mathbb {P}(0<X\le 1)=1-p_0$. Let $\mu _X$ and $\sigma _X$ be the mean value and standard deviation of X respectively. Compute $\mathbb {E}(X|X>0)$ and $\mathbb {V}(X|X>0)$.

2.42

If random variable X is exponentially distributed with a mean value of $\mu _X$, compute $\mathbb {E}(X|X>a)$, where $a>0$ is a constant.

2.43

Recall the random variable X in Problem 2.25. Compute $\mathbb {E}(X|X>\mu +b)$.

2.44

Consider two independent random variables X and Y. Show that $\mathbb {E}(X|Y=y)=\mathbb {E}(X)$ for $\forall \text {appropriate }y$.

2.45

For a normal random variable X with a CDF of $F_X(x)$ and a PDF of $f_X(x)$, show that for two real numbers $a<b$, $\mathbb {E}(X|a\le X<b)=\mu _X-\sigma _X^2\frac{f_X(b)-f_X(a)}{F_X(b)-F_X(a)}$, where $\mu _X$ and $\sigma _X$ are the mean value and standard deviation of X respectively.

2.46

For a lognormal random variable X with a mean value of $\mu _X$ and a standard deviation of $\sigma _X$, compute $\mathbb {E}(X|X>a)$, where $a>0$ is a constant.

2.47

In Problem 2.1, suppose that the probability of $D_i$ ($i=0,1,2,3,4$) is $p_i$, and that the economic loss of the structure is $L_i$ conditional on $D_i$ for $i=0,1,2,3,4$. If each $L_i$ is a random variable with a mean value of $\mu _i$ and a standard deviation of $\sigma _i$,

(a) derive the mean value and standard deviation of the post-hazard economic loss;

(b) derive the mean value and standard deviation of the post-hazard economic loss conditional on $E_1$.

2.48

Define function $C_\theta (u,v)=[\min (u,v)]^\theta \cdot (uv)^{1-\theta }$ for $\theta \in [0,1]$. Show that $C_\theta (u,v)$ is a copula [Remark. This is called the Cuadras–Augé family of copulas.].

2.49

If the joint CDF of random variables X, Y is

$$\begin{aligned} F_{X,Y}(x,y)=\exp \left( -[\exp (-\theta x)+\exp (-\theta y)]^{\frac{1}{\theta }}\right) ,\quad \theta \ge 1 \end{aligned}$$

(2.381)

show that the copula for X and Y is

$$\begin{aligned} C_\theta (u,v)=\exp \left( -\left( (-\ln u)^\theta +(-\ln v)^\theta \right) ^{\frac{1}{\theta }}\right) \end{aligned}$$

(2.382)

2.50

Consider the following two generators defined within [0, 1], (1) $\varphi (x)=\ln \left( 1-\theta \ln x\right)$ with $0<\theta \le 1$; (2) $\varphi (x)=\mathrm {e}^{1/x}-\mathrm {e}$. Find the corresponding copula functions.

2.51

The generator of the Clayton family copula is $\varphi _\theta (x)=\frac{1}{\theta }(x^{-\theta }-1)$ for $\theta \ge -1$.

(1) Derive the corresponding copula function;

(2) What is the Kendall’s tau?

2.52

Recall the generator in Problem 2.51. By referring to Eq. (2.353), what is the copula function when extended into n-dimension ($n\ge 2$)?