MATH 180A Lecture 11

Example. A weighted coin lands heads 60% of the time. I flip the coin repeatedly. What is the probability that it lands tails 5 or fewer times before it lands heads?

N = # of flips until 1st heads.


reindex

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Section 4.4

Rare Events & the Poisson Distribution

A factory produces 10 ipads per day. On average, 3 of the ipads they produce are defective and are not distributed for sale. Assuming that the probabilities of the ipads being defective are mutually independent, what is the probability that exactly 2 ipads produced are defective in a day?

= # of defective ipads in a given day.

The factory boosts production to 100 ipads per day while maintaining an average of 3 defective ipads per day. Now what is the probability that exactly 2 ipads produced are defective in a day?

= # of defective ipads in a given day

The probabilities in the previous example are converging to something… Suppose where . Then for each ,

interest rate compounding times per year, gain rate after year. Take get continuous interest

In general, we say that is a Poisson random variable w/ parameters if for all and we write . Its distribution is called the Poisson distribution.

Takeaway: A random variable counts total number of successes of “many” trials, where probability of success in each trial is very small. In other words, counts the number of occurrences of a “rare event”.

Let’s check that is a pmf:

Taylor series for centered at .

Example. Suppose . Approximate .

well approximated by