# Tutorial 1a – Analytical solution

The problem introduced in Tutorial 1 has an analytical solution [1]: the sum of exponentially distributed random variables follows a Gamma distribution with shape parameter and scale parameter . Thus, the probability of failure equals the cumulative distribution of the mentioned Gamma distribution evaluated at position .

## Step by Step Instruction

We start by defining two scalar variables named N and la.

Fesslix parameter file
const N = 100;   # number of random variables used in the problem
const la = 1;    # parameter of the exponential distribution

Note: the keyword const defines a scalar variable. The value of such a variable can be changed (i.e., this is not a constant).

Before we actually compute the analytical solution, we output a string:

Fesslix parameter file

echo "The actual probability of failure of the problem at hand is:";

Finally, we evaluate the CDF of the gamma distribution at position using the object funplot.

Fesslix parameter file
funplot cdf(  # evaluate the CDF of a distribution
60,        # at this position
gamma,     # the type of the distribution is 'gamma'
k=N,       # the shape parameter
lambda=la  # the scale parameter
);

This should return 1.481528e-6 as result.

## References

• [1] Papaioannou I., Betz W., Zwirglmaier K., Straub D. (2015): MCMC Algorithms for Subset Simulation. Probabilistic Engineering Mechanics, 41, 89-103. doi:10.1016/j.probengmech.2015.06.006

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