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Lecture 11

Lecture 11. Binomial and Poisson Distribution. Objectives. Explain Binomial distribution Calculate a binomial probability Explain Poisson Distribution Calculate a Poisson probability. Binomial Formula.

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Lecture 11

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  1. Lecture 11 Binomial and Poisson Distribution

  2. Objectives • Explain Binomial distribution • Calculate a binomial probability • Explain Poisson Distribution • Calculate a Poisson probability

  3. Binomial Formula • In a single trial, we have a simple probability model where we can denote p to be the probability of a success and (1-p)=q the probability of a failure. • If x is the successful outcome then we can write P(x)=p

  4. We note that the table shows a complete listing of all possible outcomes, hence their probabilities add up to one. Let us look at each outcome at a time. For x=0, the probability of three tails turning up can be obtained by using the multiplication rule:

  5. P(x=0)=(1-p)(1-p)(1-p) =1/2 x ½ x1/2 = 1/8 For x=1, there are three possible outcomes HTT THT TTH

  6. Again their individual probabilities can be worked out

  7. Therefore, the summation of these probabilities should give the probability of getting x=1. P(x=1)=1/8 +1/8 + 1/8 =3/8 For x =2, there are again three possible outcomes: HHT HTH THH

  8. And they add up to probability of getting x=2 which should also be 3/8. Similarly, for x=3, the probability of only one possible outcome, HHH, is P(x=3)= (p)(p)(p)=(p)3=(1/2)3=1/8

  9. We note that in x=1, the probability is made up of the two combinations of probabilities of one head and two tails. If we find out the number of combinations, the we can simplify the working.

  10. Shapes of Binomial Distribution • The shapes of a binomial distribution are determined by 2 parameters, n and p. When p= 0.5, the distribution is symmetrical. When p≠q, then the distribution will be skewed to the left if p>q, and skewed to the right if p<q. However when the value of n increases, the distribution will approach symmetry irrespective of value p.

  11. Mean and Standard Deviation of Binomial Distribution • The binomial distribution has a mean (µ) and a standard deviation (σ) which are represented by the following formulae:

  12. Poisson Distribution • Another important discrete distribution is the Poisson distribution which is particularly useful in applying to processes that can be expressed in per unit of time or space such as in queuing or wanting problems, The number of telephone calls per minute, the number of patients coming into a clinic per hour, and the number of defective items per batch can all be described by a Poisson process.

  13. The characteristics underlying a Poisson distribution are: • For a small enough unit of time or space, the probability of more than one success is practically zero; • Expected number of successes,µ, remains unchanged overtime; • The number of successes in different periods are statistically independent.

  14. The Poisson function is given by the formula:

  15. Example • The average number of patients coming into a clinic is 15 per hour. If the number of patients coming in follows a Poisson distribution, what is the probability that for any given hour, there will be exactly 10 patients?

  16. Solution • We are given

  17. Recap • Explain Binomial distribution • Calculate a binomial probability • Explain Poisson Distribution • Calculate a Poisson probability

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