Limit Theorems

A clinic is equally likely to have 2, 3, or 4 doctors volunteer for service on a given day. No matter how many volunteer doctors there are on a given day, the numbers of patients seen by these doctors are independent Poisson random variables with a mean of $30$. Let X denote the number of patients seen in the clinic on a given day.

(a) Find $E\left[X\right]$

(b) Find Var$\left(X\right)$

(c) Use a table of the standard normal probability distribution to approximate P$P\{X>65\}$.

The strong law of large numbers states that with probability 1, the successive arithmetic averages of a sequence of independent and identically distributed random variables converge to their common mean . What do the successive geometric averages converge to? That is, what is 

$\underset{n\to \infty }{\lim }{\left(\prod _{i=1}^n{X}_i\right)}^{1/n}$

Each new book donated to a library must be processed. Suppose that the time it takes to process a book has a mean of $10$ minutes and a standard deviation of $3$ minutes. If a librarian has $40$  books to process,

(a) approximate the probability that it will take more than $420$ minutes to process all these books;

(b) approximate the probability that at least $25$ books will be processed in the first $240$ minutes. What assumptions have you made? 

8.8. On each bet, a gambler loses $1$ with probability$.7$, loses $2$with probability $.2$, or wins $10$with probability $.1$. Approximate the probability that the gambler will be losing after his first $100$ bets.

Determine $t$ so that the probability that the repair person in Self-Test Problem 8.7 finishes the $20$ jobs within time t is approximately equal to $.95$.