If given:
- an arbitrary value $c>0$
- a value $b$ drawn uniformly at random between 0 and w (with $w>0$)
- and another value z drawn from the standard normal distribution
What would be the probability that $b+c*z < w$ ?
I know the probability of b being smaller than a given value $r$ is $\int_{0}^{r} \frac{1}{w}$ (so just $\frac{r}{w}$)
The probability of $c*z < w$ is $ \int_{t=0}^{w/c}f_p(t) $ with $f_p$ the probability density function of the standard normal distribution.
But I'm struggling on how to put the two together.
EDIT:
we have $g(x)=1/W$ the probability density function of b.
So using convolution: $\int_{0}^{w}f_p(t)*g(w-t) = \int_{0}^{w}f_p(t)*(1-\frac{t}{w})$