How to determine the probability of a run of normally distributed events? I'm having a random variable that takes on a normally distributed value at each month of the year. The mean and the standard deviation of the distribution is different each month (it's solar power production).
For a given year this results in a run of 12 actual values. What I'd like to do is to determine the probability of a given run occurring (say, in every month for a year the production is at the lower end of the range).
How should I do this? What I've tried is taking the CDF for each monthly distribution, then for production p at that month calculating r = CDF(p+d)-CDF(p-d) where d determines an arbitrary range around p, then taking r1 * r2 * ... * r12 as the probability of the run. But my issue is that as I'm taking the limit d --> 0 to have a more precise estimation of the probability of the run this whole thing goes to zero and that feels fishy. So this is where I'm stuck. How to do this right?

I think you've done the correct thing in setting a range around $p$, and it makes sense that as d approaches 0, your probability goes to 0. Due to the normal distribution being continuous, the probability of any exact value is 0, which is why your range approach is appropriate for a normally distributed variable.

*

*If $x_i$ ~ N($\mu_i, \sigma_i$), and $x_1$,...$x_{12}$ are independent

*$x_i$ is production for month $i$

*$\mu_i, \sigma_i$ are mean and standard deviation for month $i$

*($p-d,p+d$) defines your range

The probability of the sequence (of ranges) is:
$\Pi^{12}_{i=1}\Phi(x_i-d,x_i+d)$ = $\Pi^{12}_{i=1}(P(x_i < (x_i+d))-P(x_i < (x_i-d) ) )$
There is some limited interpretability of this result. The likelihood of most 12 event sequences here would be pretty small depending on the size of $d$ and $\sigma_i$. For example, if each $p_i$ = $\mu_i$ and $d = 2\sigma_i$, (a probability of about 0.95 of this range occurring individually each month), the probability of a sequence of 12 for this scenario is 0.54, so I'm sure you can infer how small the probabilities would get for more meaningful ranges.
