1. Estimating probability of failure
The reliability index.
Monte Carlo simulation.
Uncertainty in Monte Carlo simulations.
The uncertainty in the number of rare occurrence is the square root of the observed number.

2. Limit state function
Simulation to predict the behavior of a physical system often try to answer what is the probability of a catastrophic event.
We will use the term “probability of failure.”
Following the textbook by Choi, Grandhi, and Canfield (CGC) we will denote the behavioral quantity that defines system failure by S(X), where X is a random vector of uncertainties in the simulation.
The limit on S, or the resistance of the system is denoted as R(X). The limit state function is g=R-S.
Failure occurs when g=R-S<0.
Unfortunately an alternate definition is system response is denoted as R and resistance by C for capacity.
The material in this lecture uses Chapters 3 and 4 of Reliability-based Structural Design by Choi, Grandhi and Canfield (Springer, 2007).

3. Probability of failure and reliability index
The probability of failure is
Another way to measure safety is by the number of standard deviations the mean of g is away from the failure boundary.
This is the reliability index

4. Monte Carlo Simulation
SOURCE:
Given a random variable X and
a limit state function g(X):
sample X: [x1,x2,…,xn];
Calculate [g(x1),g(x2),…,g(xn)]; use to approximate statistics of g.
Example: X is U[0,1]. Use MCS to find mean of X2
x=rand(10); y=x.^2; %generates 10x10 random matrix
sumy=sum(y)/10
sumy =
sum(sumy)/ ans =0.3580
What is the true mean
SOURCE:

5. Evaluating probabilities of failure
Failure is defined in terms of a limit state function where failure occurs when g(X)<0, where X is a vector of random variables.
Probability of failure is estimated as the ratio of number of negative g’s, m, to total MC sample size, N
The accuracy of the estimate is poor unless N is much larger than 1/Pf
For small Pf

6. Example Estimate the probability that x=N(0,1)>1
x=randn(1,1000); x1=0.5*(sign(x-1)+1); pf=sum(x1)/1000.; pf =0.1550
Repeating the process obtained: 0.136, 0.159, 0.160, 0.172, 0.154,
Exact value
In general, for 10% accuracy of probability you need 100 failed samples.

7. Top Hat question
Sampling a distribution with 10,000 points, the mean of the sample was 6, the standard deviation of the sample was 2, and 100 points were negative. Estimate the noise (standard deviation) in the mean and number of negative points over repeated 10,000 samples.
0.02, 10
0.2,1
0.02,1
0.2,10