Difficult Beta Distribution Problem .

Let [math]X,Y[/math] be independent [math]\Gamma(2,a)[/math]-distributed variables. What is the conditional distribution of [math]X[/math] given that [math]X + Y = 2[/math]?

Let’s solve a more general version of this in two ways: by representation and by densities. Let [math]X,Y[/math] be i.i.d. Gamma([math]b,\lambda[/math]), where [math]\lambda[/math] is the scale parameter. Let’s find the conditional distribution of [math]X|(X+Y=c)[/math], for any positive constant [math]c[/math].Representation: One of the most important properties of the Gamma distribution is its symbiotic relationship with the Beta, as discussed at What is an intuitive explanation of this relationship between a Beta and Gamma distribution? X∼Gamma(a,λ)[math]X∼Gamma⁡(a,λ)[/math]  Y∼Gamma(b,λ)[math]Y∼Gamma⁡(b,λ)[/math]  X⊥⊥Y[math]X⊥⊥Y[/math] implies...XX+Y∼Beta(a,b)[math]XX+Y∼Beta⁡(a,b)[/math] XX+Y⊥⊥X+Y This is a representation for a Beta r.v. in terms of Gamma r.v.s.Using this relationship, we can solve the problem in one line:[math]X|(X+Y=c) \sim \frac{X}{X+Y} \cdot (X+Y) | (X+Y = c) \sim c \cdot \frac{X}{X+Y} \sim c \cdot \textrm{Beta}(b,b)[/math]That is, the conditional distribution is a Beta([math]b,b[/math]), rescaled to be between [math]0[/math] and [math]c [/math] (note that the support make sense, since [math]X,Y[/math] are positive so one of them can’t exceed their sum). A key part of the above argument is the fact that [math]X/(X+Y)[/math] is independent of [math]X+Y[/math], so we can plug in the information and then drop the conditioning.Densities: Now let’s work with the relevant densities. I will use George Foreman notation for brevity. Letting [math]T=X+Y[/math], have[math]f(x|t) = \frac{f(x,t)}{f(t)} = \frac{f(x)f(t-x)}{f(t)}[/math]for [math]0

What classes of models can be used to predict time series distributions?

Ironically enough, I am in the midst of negotiating a possible contract where I would do this very thing. It’s a fairly difficult problem, depending on the business and how much client data exists. For me, it’s a variation on a theme of various ideas I have been working on since late undergraduate, through employment and to this day. And I still have no idea if it's possible to solve in a general case.The real issue is the fact that we are considering, quite possibly, individuals. Usually, when we model, individuals vary - some who spend more offset those who spend less. It's a very familiar variation on regression towards the mean.Prediction of individuals over customer lifetimes is easy - easier, I should say. Individuals are inherently irrational on many levels, however, and prediction of when (much less why) they are going to do something tends to be harder. Often a lot harder. If the individuals in question are making purchases for a specific cause - say, they are in business for themselves and their business is relatively predictable (because it depends on multiple other individuals) - then this problem may be possible to solve. If we are a retailer trying to get probabilities of individual buying patterns - this is a lot harder.