NProbability not reliability analysis?

Question

I have 2 random independent variables x and y. x is the strength of a bar and y is the force applied to the bar. x has a Lognormal distribution with mean 10. and standard deviation 2. y has a Normal distribution with mean 5. and standard deviation 2. The bar will fail if we have x-y = 0. How to calculate the probability of the bar to fail? I tried the following and expected that probability of fail would be 0.032.

NProbability[
 x - y == 0.,
 {
  x \[Distributed] LogNormalDistribution[10., 2.],
  y \[Distributed] NormalDistribution[5., 2.]
  },
 Method -> {"MonteCarlo"}
 ]

Answer 1

You have two related problems. The first is that NProbability finds the probability of a condition being met. However, your condition x - y == 0 is only true when x and y are exactly equal, which happens with probability zero.

Fixing the condition to y > x (and using the default Method), we get a probability of:

NProbability[
 y > x,
 {
  x \[Distributed] LogNormalDistribution[10, 2], 
  y \[Distributed] NormalDistribution[5, 2]
 },
 AccuracyGoal -> 30
]

(* 0. *)

Practically zero, much lower than you expect. I think you have a second problem, which is that LogNormalDistribution does not do what you expect it to. It is the distribution of Exp[x], where x is normally distributed with the parameters given to LogNormalDistribution. Therefore in general the mean and standard deviation of the distribution will not be the numbers you put in.

We can use Solve to find out what the parameters should be:

soln = NSolve[Through[{Mean, StandardDeviation}[LogNormalDistribution[a, b]]] == {10, 2}
 && b > 0, {a, b}, Reals]

(* {{a -> 2.28297, b -> 0.198042}} *)

Now we can use NProbability again (without any special options):

NProbability[y > x, {
 x \[Distributed] LogNormalDistribution[a, b] /. First@soln,
 y \[Distributed] NormalDistribution[5, 2]
}]

(* 0.0321748 *)

And this time we get exactly what you expect.