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STT 315

STT 315. Ashwini Maurya. This lecture is based on Chapters 4.3-4.4. Acknowledgement: Author is thankful to Dr. Ashok Sinha, Dr . Jennifer Kaplan and Dr. Parthanil Roy for allowing him to use/edit some of their slides. Binomial random variable.

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STT 315

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  1. STT 315 Ashwini Maurya This lecture is based on Chapters 4.3-4.4 Acknowledgement: Author is thankful to Dr. Ashok Sinha, Dr. Jennifer Kaplan and Dr. Parthanil Roy for allowing him to use/edit some of their slides.

  2. Binomial random variable • The experiment consists of n independent and identical trials. • Each trial has only two outcomes (that is why it is also called binary trial). • One outcome is called the success and the other one is called the failure. • P(success) = p, and it remains fixed throughout the experiment. P(failure) = 1-p =:q. • Let X be the number successes in n trials. X is a binomial random variable and we write • The probability distribution function of X is if

  3. Binomial probability distribution • The probability distribution function of X is if • p = P(success), • q =1-p, • n = number of trials • x = number of successes in n trials • where • .

  4. TI 83/84 Plus commands Suppose and we want to compute • Press [2nd] & [VARS] (i.e. [DISTR]). • Under DISTRselect 0: binompdf(and press ENTER. • On screen binompdf( will appear. • Type the values of n,p,x) and press ENTER. Suppose and we want to compute • Press [2nd] & [VARS] (i.e. [DISTR]). • Under DISTRselect A: binomcdf(and press ENTER. • On screen binomcdf( will appear. • Type the values of n,p,x) and press ENTER.

  5. Example Suppose a fair die is thrown 30 times. • What is the probability that we get exactly 6 sixes? Suppose X denote the number of sixes in 30 throws. Then Thus • What is the probability that we get at most 4 sixes?

  6. Example Suppose a fair die is thrown 30 times. • What is the probability of getting more than 7 sixes? Suppose X denote the number of sixes in 30 throws. Then Therefore • What is the probability that we get at least 7 sixes?

  7. Example Suppose a fair die is thrown 30 times. • What is the expected number of sixes? Suppose X denote the number of sixes in 30 throws. Then • What is the standard deviation of number of sixes?

  8. Example Suppose a fair die is thrown 30 times. • What is the probability that number of sixes appears is within two standard deviation of the average? Suppose X denote the number of sixes in 30 throws. Then Here Hence

  9. Poisson random variable • A discrete random variable X is said to be a Poisson random variable if its probability distribution is where and is a constant. • Notation: ). • .

  10. TI 83/84 Plus commands Suppose and we want to compute • Press [2nd] & [VARS] (i.e. [DISTR]). • Under DISTRselect B: poissonpdf(and press ENTER. • On screen poissonpdf( will appear. • Type the values of ,x) and press ENTER. Suppose and we want to compute • Press [2nd] & [VARS] (i.e. [DISTR]). • Under DISTRselect C: poissoncdf(and press ENTER. • On screen poissoncdf( will appear. • Type the values of ,x) and press ENTER.

  11. Example Suppose that the number of defective items produced by a machine follows Poisson distribution with average 1.7 defective items per shift. We select a shift at random. What is the probability that in that shift the machine producedexactly 4 defective items? Suppose X denote the number of defective items produced by the machine in that shift. Then So

  12. Example Suppose that the number of defective items produced by a machine follows Poisson distribution with average 1.7 defective items per shift. We select a shift at random. What is the probability that in that shift the machine producedat most 3 defective items? Suppose X denote the number of defective items produced by the machine in that shift. Then Hence

  13. Example Suppose that the number of defective items produced by a machine follows Poisson distribution with average 1.7 defective items per shift. What is the standard deviation of defective items produced per shift? Suppose X denote the number of defective items produced by the machine in that shift. Then Hence

  14. Example Suppose that the number of defective items produced by a machine follows Poisson distribution with average 1.7 defective items per shift. We select a shift at random. What is the probability that in that shift number of defective items produced by the machine is within one standard deviation of the average? Here Hence

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