Probability of failure

Question

A structure will fail if subjected to a load greater then its own resistance:

failure := load > resistance

We can assume that the load and the resistance are independent.

By means of probability density functions (pdf) and cumulative density functions (cdf) of the load and of the resistance, is it correct to say that the probability of failure can be calculated by: $$ p_{failure} = \int_{-\infty}^\infty PDF_{load} * CDF_{resistance} $$ ?

I'm trying to teach myself statistics, but sometimes without a good reference it's difficult to know what to look for. What this "tool" would be called by the way? I'm sure there's a proper name for this.

Answer 1

Let $X$ denote the resistance and $Y$ the load. Then, \begin{align} P\{Y > X\} &= \int_{y=-\infty}^\infty \int_{x=-\infty}^y f_{X,Y}(x,y) \,\mathrm dx \,\mathrm dy\\ &= \int_{y=-\infty}^\infty \int_{x=-\infty}^y f_{X}(x)f_{Y}(y) \,\mathrm dx \,\mathrm dy & \scriptstyle{\text{because}~X~\text{and} ~Y~\text{are independent}}\\ &= \int_{y=-\infty}^\infty f_{Y}(y)\left[ \int_{x=-\infty}^y f_{X}(x) \,\mathrm dx\right] \,\mathrm dy\\ &= \int_{y=-\infty}^\infty f_{Y}(y)F_{X}(y) \,\mathrm dy\\ \text{that is}, \qquad p_{failure} &= \int_{-\infty}^\infty PDF_{load} \times CDF_{resistance} \end{align} which is the formula that you are asking about, without needing to worry about convolutions, cross-correlations, complex numbers, and the like as in Sean Easter's answer.

As a practical matter, $X$ and $Y$ are likely to take on nonnegative values only, in which case the above integral need only be on the positive real line.

Answer 2

Rephrased, the probability of failure is equivalent to the probability that resistance - load is less than zero. What you're looking for is the distribution of the difference of random variables.

Since these are independent, you can use convolution to solve for their difference. But it's applied to the densities, not a cumulative density. Also, the convolution is itself an infinite integral. Let $X$ represent load, $Y$ resistance. You'd want to convolve $p_{X}(-t)$ and $p_{Y}(t)$, called the cross-correlation in signal processing:

$$p_{Y-X}(\tau) = p_x(-\tau) \ast p_Y(\tau)= \int_{-\infty}^{\infty}p_{X}(t)p_{Y}(\tau + t)dt$$

Strictly, cross-correlation is equivalent to the convolution of $p_X^*(-\tau)$ and $p_Y(\tau)$, where the asterisk is the complex conjugate. Since densities are real-valued, $p_X^*(-\tau) = p_X(-\tau)$ and there's no need to worry.

The probability of failure is the probability that the difference is less than zero, which you can find by integrating the density of the differences up to zero: $\int_{-\infty}^0p_{Y-X}(\tau)d\tau$. (I.e., the CDF of the difference.) You can do all of this numerically, but the more you can do analytically, the more efficient it will be.