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?
 A: You want to know the likelihood that the solar power production falls within a particular specified range every month. You have correctly observed that if that range is defined to be small, the likelihood of falling within that range is also small, while if that range is defined to be large, the likelihood is also large, which makes complete sense. If you set the range very wide, you're effectively asking the likelihood that the power production falls somewhere between zero and the maximum possible output of the power plant - with a wide enough range, you're guaranteed to find power production in that range. Conversely, if you set the range very narrow, you're looking for the probability that the power production each month is within a milli-Watt (or some other small unit) of some specified value, which is very unlikely to occur.
No setting of the range is inherently useful, you need to figure out exactly what question you want to answer. Perhaps you want to know the likelihood that the power output falls within 1% of the mean output for that month. Maybe that range is unrealistically narrow, and you ought to be looking within 10% instead. You could also define the range based on the distribution itself - you could look at the likelihood of falling within the bottom 50% of observed values, which by definition would be 50%. In this case, the likelihood of having below-median power production for N months would simply be $0.5 ^N$, assuming independence of power production from month to month. Perhaps you want to know the likelihood of having power production in the bottom 10% per month at any point in the year, which would be yet another calculation.
You need to define what are the ranges you actually care about in order for the calculated probability have a useful interpretation - stating that the power production definitely is greater than zero and less than the plant's total possible output is not interesting in the slightest, nor is the statement that the power production is almost certainly not exactly 10.7346597 MW, although both of those are completely valid probabilistic analyses.
You'll need some domain knowledge here, perhaps there is a "normal operating range" for the plant where higher or lower output indicates some problem - that might represent a "useful" range to use in a probability calculation, which would allow you to make statements about the likelihood of out-of-range, abnormal behavior over the course of a year. But without that domain knowledge, it's impossible to know whether looking at variation on the order of milli-Watts or giga-Watts is actually useful.
A: 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.
