Probability with intersecting normal distributions

Question

There are two independent random variables $a$ and $b$, each distributed normally with their own parameters. Given the means and standard deviations for $a$ and $b$, how can I calculate $P(a < b)$?

Answer 1

This is easy if you look for the probability distribution of $a-b$.

As $a \sim \mathcal{N}(\mu_a, \sigma_a^2)$ and $b\sim \mathcal{N}(\mu_b, \sigma_b^2)$ are independant, you have that

$$a-b = Z \sim \mathcal{N}(\mu_a-\mu_b, \sigma_a^2+\sigma_b^2)$$

Then you can calculate $P(Z<0)$ as usual