I am modeling the probability of generating a sentence of dictionary words with a probability distribution over letter frequencies and word lengths.
For instance, a 10 word sentence is expected to have 2 two letter words, 2 three letter words, 2 four letter words, 1 five letter word, etc.
Using the letter frequencies, the two letter words have a 6.6% chance of being in the dictionary, the three letter words have a 2.3% chance, etc.
So, to calculate the probability that a 10 word sentence has all dictionary words I find the probability of all the k letter words being in the dictionary, and multiply these probabilities together: $0.066^2 \times 0.023^2 \times ... = 8.4 \times 10^{-33}$
Now, I can generalize this by turning it into a probability that a single word is in the dictionary when generated from the letter and length frequency distributions. $P[w \in Dict | Ltr, Len] = \sum P[w_k \in Dict | Ltr]^{P[w \in Len_k]}$.
However, this notation looks weird. I have not seen one distribution exponentiated by another distribution before. Is there a standard notation for what I am trying to do here?