Question

Let \(A_{0}, A_{1}, \ldots, A_{r}\), be \(r+1\) events which can oceur as outcomes of an experiment. Let \(p_{l}\) be the probability of the occurrence of \(A_{l}(i=0,1,2, \ldots, r)\). Suppose we perform independent trials until the event \(A_{0}\) occurs \(k\) times. Let \(X_{i}\) be the number of occurrences of the event \(A_{i} .\) Show that \(\operatorname{Pr}\left\\{X_{1}=x_{1}, \ldots, X_{r}=x_{r} ; A_{0}\right.\) oceurs for the \(k\) th time at the \(\left(k+\sum_{i=1}^{r} x_{i}\right)\) th trial \(\\}\) $$ =\frac{\Gamma\left(k+\sum_{i=1}^{r} x_{i}\right)}{\Gamma(k) \prod_{i=1}^{r} x_{i} !} p_{0}^{k} \prod_{i=1}^{r} p_{i^{i}}^{x_{i}} $$

Short answer

In conclusion, the probability that \(X_1=x_1\), \(X_2=x_2\), ..., \(X_r=x_r\) and \(A_0\) occurs for the \(k\)th time at the \((k+\sum_{i=1}^{r} x_i)\)th trial is given by: $$ \operatorname{Pr}\left\{X_{1}=x_{1}, \ldots, X_{r}=x_{r} ; A_{0}\right\} =\frac{\Gamma\left(k+\sum_{i=1}^{r} x_{i}\right)}{\Gamma(k) \prod_{i=1}^{r} x_{i} !} p_{0}^{k} \prod_{i=1}^{r} p_{i^{i}}^{x_{i}} $$ This expression allows us to compute the probability of specific occurrences of each event under the given conditions.

Step-by-step solution

Define the Notation

: We are given \(r+1\) events, \(A_0\), \(A_1\), ..., \(A_r\), where \(p_i\) is the probability of the occurrence of event \(A_i\). We perform independent trials until event \(A_0\) occurs \(k\) times. Let \(X_i\) be the number of occurrences of event \(A_i\) up to that point.

Set up the Expression for the Probability

: We are interested in finding the probability that \(X_1=x_1\), \(X_2=x_2\), ..., \(X_r=x_r\), and that \(A_0\) occurs for the \(k\)th time at the \((k+\sum_{i=1}^{r} x_i)\)th trial.

Simplify and Expand the Expression

: We are given the following formula for the probability: $$ \operatorname{Pr}\left\\{X_{1}=x_{1}, \ldots, X_{r}=x_{r} ; A_{0}\right.$ occurs for the \(k\) th time at the \(\left(k+\sum_{i=1}^{r} x_{i}\right)\) th trial \(\\}\) $$ =\frac{\Gamma\left(k+\sum_{i=1}^{r} x_{i}\right)}{\Gamma(k) \prod_{i=1}^{r} x_{i} !} p_{0}^{k} \prod_{i=1}^{r} p_{i^{i}}^{x_{i}} $$

Apply the Given Formula to Calculate the Probability

: To calculate the probability, we simply plug in the given values for \(k\), \(x_1\), \(x_2\), ..., \(x_r\), \(p_0\), \(p_1\), ..., \(p_r\) into the formula: $$ \operatorname{Pr}\left\\{X_{1}=x_{1}, \ldots, X_{r}=x_{r} ; A_{0}\right.$ occurs for the \(k\) th time at the \(\left(k+\sum_{i=1}^{r} x_{i}\right)\) th trial \(\\}\) $$ =\frac{\Gamma\left(k+\sum_{i=1}^{r} x_{i}\right)}{\Gamma(k) \prod_{i=1}^{r} x_{i} !} p_{0}^{k} \prod_{i=1}^{r} p_{i^{i}}^{x_{i}} $$ The result is the probability we are looking for, which shows the relationship between the occurrences of each event and the probabilities provided.