1. MA305 Binomial Distribution and Poisson Distribution By: Prof. Nutan Patel Asst. Professor in Mathematics IT-NU A-203 patelnutan.wordpress.com MA305 Mathematics for ICE1

2. Binomial Distribution MA305 Mathematics for ICE2

3. Ex. Team A has probability 2/3 of winning whenever it plays. If A plays 4 games, find the probability that A wins (i) exactly 2 games (ii) at least 1 game (iii) more than half of the games. Ans: n=4, p=2/3, q=1/3. i. P(2)=8/27. ii. 80/81. iii. P(3)+P(4)=16/27. MA305 Mathematics for ICE3

4. Ex: In sampling a large number of parts manufactured by a machine, the mean number of defectives in a sample of 20 is 2. out of 1000 such samples, how many would be expected to contain at least 3 defective parts? Ans: n=20, p=0.1, P(X≥3)=1-{ P(0) + P(1) + P(2) } = 1- { 0.12157 + 0.27017 + 0.28517 } = 0.323. Expected number=1000*0.323=323. MA305 Mathematics for ICE4

5. 5 XP(X=x)P(X) 0 0.000244 1 0.004395 2 0.032959 3 0.131836 4 0.296631 5 0.355957 6 0.177979

6. Poisson Distribution MA305 Mathematics for ICE6

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8. Application of Poisson Distribution Example of rare events:  Number of accidents on a highway.  Number of printing mistakes per page.  Number of deaths per day or per week due to a rare disease in a big city.  Number of defectives in a production centre.  The count of Bacteria per c.c. in blood. MA305 Mathematics for ICE8

9. Properties of Poisson Distribution MA305 Mathematics for ICE 9

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12. EX: If 10% of the rivets produced by a machine are defective, find the probability that out of 5 rivets chosen at random (i) none will be defective, (ii) one will be defective, and (iii) at least two will be difective. Ans: 0.5905, 0.32805, 0.08146. MA305 Mathematics for ICE12

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15. Graphical Comparison for Binomial and Poisson Distribution Ex: For n=100, p=0.04, so, =4. MA305 Mathematics for ICE15 xB.D. P(x)P.D. P(x) 00.016870.018316 10.0702930.073263 20.1449790.146525 30.1973330.195367 40.1993880.195367 50.1595110.156293 60.1052330.104196 70.058880.05954 80.028520.02977 90.0121470.013231 100.0046060.005292