I'm really bad at calculus and math and sometimes struggle with the posterior math we encounter on Bayesian credibility questions.

So can someone verify, is this all true? Are there other situations beyond these that anyone has seen in their studies?

Let X be the counting variable, and let X's distribution have a parameter Y which has its own continuous probability distribution where Y > 0. Then these are the 4 types of situations that are likely to show up on exam since they were in the sample questions:

Situation 1: "solve the probability X is less than some number,"

**but no information given on what values X has taken previously**:

in which F(X|Y) is the conditional CDF of X given Y, and f(y) is the prior distribution of y

Situation 2: "solve for the probability Y is between a and b,"

**no information given on what value X has been previously**:

In which Fy is the CDF of Y

Situation 3: From sample #43 and #157,

**given X was some number z **previously, what's the probability parameter Y is between a and b is:

in which f(y|x=z) is the posterior distribution

Situation 4: From sample #76, given

**X was some number z**, whats the probability X is between a and b next period is:

and we would replace the probability that X is between a and b given y, with the probability of the event of whatever information we were asked to solve for for X possibly being next period.