# Calculate $\mathbb{P}[Y=y|X=x]$ where $X$=# claims reported diring firs year, $Y$=# claims that will eventually be reported

A property-casualty insurance company issues automobile policies on a calendar year basis only. Let $$X$$ be a random variable representing the number of accident claims reported during calendar year 2005 on policies issued during calendar year 2005. Let $$Y$$ be a random variable representing the total number of accident claims that will eventually be reported on policies issued during calendar year 2005. The probability that an individual accident claim on a 2005 policy is reported during calendar year 2005 is $$d$$. Assume that the reporting times of individual claims are mutually independent. Assume also that $$Y$$ has the negative binomial distribution, with fixed parameters $$r$$ and $$p$$, given by $$\mathbb{P}[Y=y]=\binom{r+y-1}{y}p^{r}(1-p)^{y}$$ for $$y=0,1,\ldots$$. Calculate $$\mathbb{P}[Y=y|X=x]$$ the probability that the total number of claims reported on 2005 policies is $$y$$, given that $$x$$ claims have been reported by the end of the calendar year.

Remark: I know that the solution requires the use of Bayes' Theorem and Theorem of Total Probability, and the identity $$\binom{y}{x}\binom{r+y-1}{y}=\binom{r+x-1}{x}\binom{(r+x)+(y-x)-1}{y-x}$$.

I have not been able to correctly describe $$X$$ or include $$d$$ in the analysis. I need your help to understand this better.

1: Find the conditional distribution of $$X$$ conditional on $$Y$$, $$\mathbb{P}[X|Y]$$, which involves $$d$$.
2: Get joint distribution of $$X$$ and $$Y$$, $$\mathbb{P}[X,Y] = \mathbb{P}[Y]\mathbb{P}[X|Y]$$.
1. Get marginal distribution of X, $$\mathbb{P}[X] = \sum _Y\mathbb{P}[X,Y]$$.
2. Your answer: Conditional distribution of $$Y$$ given $$X$$,$$\mathbb{P}[Y|X] = \frac {\mathbb{P}[X,Y]}{\mathbb{P}[X]}$$.