# Calculating the probability of an inequality with two random variables

I am analyzing a timing circuit I designed, and I need to calculate the probability of a certain event (bit error). For example, I have derived this equation:

$(1 + x) / d < 1 / M$,

where x is a random variable with a normal distribution, d is a random variable with uniform distribution, and M is a constant.

I would like to be able to calculate the probability that this inequality is true, for a given distribution for x, a given distribution for d, and a given M. For example, let's say x has a normal distribution with mean = 0 and SD = 0.05, d has a uniform distribution between 0 and 1, and M = 3. How would I calculate the probability that the inequality is true?

If there was a single random variable, I could do it. But I am lost and don't know how to solve it. I tried searching and reading a statistics textbook, but I don't know what to look for.

• Can we assume that the support of $d$ is always included in $\mathbb R^{\ge 0}$ ? – Elvis Nov 29 '13 at 8:27
• @Elvis, yes, you can assume that. I think. – travisbartley Nov 29 '13 at 8:31

It should be possible to derive the probability $<1/M$ from there. Another option perhaps is bootstrapping, where you would take random samples from your data and evaluate the criterion. The proportion of cases in which it is true would be the probability of your event.