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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.

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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 $X$,$\mathbb{P}[Y|X] = \frac {\mathbb{P}[X,Y]}{\mathbb{P}[X]}$.

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