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- 1. Binomial Distribution and Applications
- 2. Binomial Probability Distribution Is the binomial distribution is a continuous distribution?Why? Notation: X ~ B(n,p) There are 4 conditions need to be satisfied for a binomial experiment: 1. There is a fixed number of n trials carried out. 2. The outcome of a given trial is either a “success” or “failure”. 3. The probability of success (p) remains constant from trial to trial. 4. The trials are independent, the outcome of a trial is not affected by the outcome of any other trial.
- 3. Comparison between binomial and normal distributions
- 4. Binomial Distribution If X ~ B(n, p), then where successof trials.insuccessesofnumberr 11!and10!also,1...)2()1(! yprobabilitP n nnnn .,...,1,0r)1( )!(! ! )1()( npp rnr n ppcrXP rnrrnr n r
- 5. Exam Question Ten percent of computer parts produced by a certain supplier are defective. What is the probability that a sample of 10 parts contains more than 3 defective ones?
- 6. Solution : Method 1(Using Binomial Formula):
- 7. Method 2(Using Binomial Table):
- 8. From table of binomial distribution :
- 9. Example 2 If X is binomially distributed with 6 trials and a probability of success equal to ¼ at each attempt. What is the probability of a)exactly 4 succes. b)at least one success.
- 10. Example 3 Jeremy sells a magazine which is produced in order to raise money for homeless people. The probability of making a sale is, independently, 0.50 for each person he approaches. Given that he approaches 12 people, find the probability that he will make: (a)2 or fewer sales; (b)exactly 4 sales; (c)more than 5 sales.
- 11. Normal Distribution
- 12. Normal Distribution In general, when we gather data, we expect to see a particular pattern to the data, called a normal distribution. A normal distribution is one where the data is evenly distributed around the mean, which when plotted as a histogram will result in a bell curve also known as a Gaussian distribution.
- 13. thus, things tend towards the mean – the closer a value is to the mean, the more you’ll see it; and the number of values on either side of the mean at any particular distance are equal or in symmetry.
- 14.
- 15. Z-score with mean and standard deviation of a set of scores which are normally distributed, we can standardize each "raw" score, x, by converting it into a z score by using the following formula on each individual score:
- 16. Example 1 a) Find the z-score corresponding to a raw score of 132 from a normal distribution with mean 100 and standard deviation 15. b) A z-score of 1.7 was found from an observation coming from a normal distribution with mean 14 and standard deviation 3. Find the raw score. Solution a)We compute 132 - z = __________ = 2.133 15 b) We have x - 1.7 = ________ 3 To solve this we just multiply both sides by the denominator 3, (1.7)(3) = x - 14 5.1 = x - 14 x = 19.1
- 17. Example 2 Find a) P(z < 2.37) b) P(z > 1.82) Solution a)We use the table. Notice the picture on the table has shaded region corresponding to the area to the left (below) a z-score. This is exactly what we want. Hence P(z < 2.37) = .9911 b) In this case, we want the area to the right of 1.82. This is not what is given in the table. We can use the identity P(z > 1.82) = 1 - P(z < 1.82) reading the table gives P(z < 1.82) = .9656 Our answer is P(z > 1.82) = 1 - .9656 = .0344
- 18. Example 3 Find P(-1.18 < z < 2.1) Solution Once again, the table does not exactly handle this type of area. However, the area between -1.18 and 2.1 is equal to the area to the left of 2.1 minus the area to the left of -1.18. That is P(-1.18 < z < 2.1) = P(z < 2.1) - P(z < -1.18) To find P(z < 2.1) we rewrite it as P(z < 2.10) and use the table to get P(z < 2.10) = .9821. The table also tells us that P(z < -1.18) = .1190 Now subtract to get P(-1.18 < z < 2.1) = .9821 - .1190 = .8631
- 19. Poisson distribution
- 20. Definitions a discrete probability distribution for the count of events that occur randomly in a given time. a discrete frequency distribution which gives the probability of a number of independent events occurring in a fixed time.
- 21. Poisson distribution only apply one formula: Where: X = the number of events λ = mean of the event per interval Where e is the constant, Euler's number (e = 2.71828...)
- 22. Example: Births rate in a hospital occur randomly at an average rate of 1.8 births per hour. What is the probability of observing 4 births in a given hour at the hospital? Assuming X = No. of births in a given hour i) Events occur randomly ii) Mean rate λ = 1.8 Using the poisson formula, we cam simply calculate the distribution. P(X = 4) =( e^-1.8)(1.8^4)/(4!) Ans: 0.0723
- 23. If the probability of an item failing is 0.001, what is the probability of 3 failing out of a population of 2000? Λ = n * p = 2000 * 0.001 = 2 Hence, use the Poisson formula X = 3, P(X = 3) = Ans: 0.1804
- 24. Example: A small life insurance company has determined that on the average it receives 6 death claims per day. Find the probability that the company receives at least seven death claims on a randomly selected day.
- 25. Analysis method 1st: analyse the given data. 2nd: label the value of x, λ At least 7 days, means the probability must be ≥ 7. but the value will be to the infinity. Hence, must apply the probability rule which is P(X ≥ 7) = 1 – P(X ≤ 6) P(X ≤ 6) means that the value of x must be from 0, 1, 2, 3, 4, 5, 6. Total them up using Poisson, then 1 subtract the answer. Ans = 0.3938
- 26. Example: The number of traffic accidents that occurs on a particular stretch of road during a month follows a Poisson distribution with a mean of 9.4. Find the probability that less than two accidents will occur on this stretch of road during a randomly selected month. P(x < 2) = P(x = 0) + P(x = 1) Ans: 0.000860

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