6.5. Designing with Probability

This chapter is closed with an illustration of how our probability models can be used in practice, specifically in the design of some object or system. As engineers making decisions under uncertain conditions (i.e., the unknown values of our random variables) we face a serious conundrum1: our design must specify a deterministic value, which will eventually be used to build the object or system of interest. In the face of uncertainty, this means there will always be a non-zero probability that the object or system can fail! Thus, the choice is not simply how robust to design somthing, but rather: how safe should it be designed. In other words, what is an acceptable probability of failure for the design?

This question can be answered by applying the concepts of risk and risk analysis, which will be covered in a later chapter. For now, we will assume that an acceptable failure probability has been determined and use our probability models to evaluate what the design value should be.

Discrete Events

Previous sections in this chapter introduced multivariate distributions and, specifically, a few approaches for incorporating dependence in our probabilistic assessment of more than one random variable of interest. Until this point, examples have been illustrated using a simple paradigm: “AND” and “OR” probabilities (analogous to intersection and union in set theory), where each random variable is assumed to have a critical or threshold case with some probability of exceedance, p, for example (if \(X_1\) has the lognormal distribution):

\[ X_1\sim \mathrm{LN}(\mu, \sigma) \;\; \rightarrow \;\; P[X_1 \leq x_{threshold}] = 1 - F_{X_1}(x_{threshold}) \]

However, for the two random variable case, this limits us to only a few relevant design situations: the various combinations where each variable is either greater than or less than a threshold value. This is very restrictive in practice. Fortunately we already have a tool at hand for evaluating more complex situations: the concept of functions of random variables!

Functions of Random Variables

Given a vector of random variables \(X\) and function \(q\), the output, \(Y\), is a function of random variables:

\[ Y = q(X) \]

While \(Y\) is itself a random variable, its distribution is known only for a few special cases, for example, if \(q\) is linear and \(X\) is the multivariate Gaussian, then \(Y\) is also Gaussian. This is no longer true if the function is non-linear or the marginal distributions of the random variables are non-Gaussian. Luckily we are usually able to empirically find the distribution of \(Y\) using Monte Carlo Sampling.

It is important also to note that although the distribution of \(Y\) is univariate, it is related to the underlying multivariate distribution of \(X\) via the function \(q\). As such, the model chosen for \(f_X(x)\) will have an impact on the distribution of \(Y\).

Tip

In some fields, for example structural reliability and related branches of civil engineering, it is common to formulate the function of random variables as a limit state function, where the threshold value of the function dividing a safe and failed state is found. The function is then reformulated as a new function, \(g\) with output \(Z\) is defined such that 0 defines the threshold state and negative values the failed state. The probability of being in the failed state, \(p_f\), is found as follows:

\[ Z = g(X) \;\; \rightarrow \;\; p_f = P[Z\leq0] \]

Regardless of the underlying multivariate distribution of the input random variables, the task at hand is to compute the design value, \(y_{design}\), for a given probability of interest, \(p_{design}\):

\[ y_{design} = F_Y^{-1}(p_{design}) \]

where \(F_Y^{-1}\) is the inverse CDF of the distribution of \(Y\). Although we rarely know the exact distribution of \(Y\), it is straightforward to compute the design value using Monte Carlo Simulation.

Next Steps

The following pages work through an example related to flooding on a river to illustrate the role that the random variable inputs of a function of random variables have on . Specifically, the following pages will:

1. Consider a univariate design case for the discharge capacity of a flood protection system on a river.

2. Consider a bivariate design case where two rivers contribute to flooding and make the distinction between AND and OR perspectives explicit.

Attribution

This chapter was written by Patricia Mares Nasarre and Robert Lanzafame. Find out more here.

---

1

a confusing and difficult problem or question (source: Oxford Languages, via Google).