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CHAPTER 1 EQT 271 (part 2)

CHAPTER 1 EQT 271 (part 2). BASIC STATISTICS. 1.4 PROBABILITY DISTRIBUTION. A probability distribution is obtained when probability values are assigned to all possible numerical values of a random variable. Probability distribution can be classified either discrete or continuous.

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CHAPTER 1 EQT 271 (part 2)

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  1. CHAPTER 1 EQT 271 (part 2) BASIC STATISTICS

  2. 1.4 PROBABILITY DISTRIBUTION • A probability distribution is obtained when probability values are assigned to all possible numerical values of a random variable. • Probability distribution can be classified either discrete or continuous.

  3. THE BINOMIAL DISTRIBUTION An experiment in which satisfied the following characteristic is called a binomial experiment: 1. The random experiment consists of n identical trials. 2. Each trial can result in one of two outcomes, which we denote by success, S or failure, F. 3. The trials are independent. 4. The probability of success is constant from trial to trial, we denote the probability of success by p and the probability of failure is equal to (1 - p) = q.

  4. Example of Binomial Distribution • A university found that 20% of its students withdraw without completing the introductory statistics course. Assume that 20 students registered for the course. Find the probability…. • In Kuala Lumpur, 30% of workers take public transportation daily. In a sample of 10 workers,…. • Suppose 20% of the marbles packed in a box are red in color. Suppose 4 marbles are chosen at random. Find the probability of …

  5. 1) Calculate probability using formula A binomial experiment consist of n identical trial with probability of success, p in each trial. The probability of x success in n trials is given by • x = 0, 1, 2, ......, n

  6. 2) Calculate probability using table • Table of binomial can be used to find the probabilities using the following rules as the guidelines.

  7. The Mean and Variance of X If X ~ B(n,p), then where • n is the total number of trials, • p is the probability of success and • qis the probability of failure.

  8. Example 1 • Given that X~B(12, 0.4), find

  9. Try to find the probabilities using table. Do you get the same answer??? solutions

  10. Exercise 1 • In Kuala Lumpur, 30% of workers take public transportation daily. In a sample of 10 workers, • What is the probability that exactly three workers take public transportation daily? • What is the probability that at least three workers take public transportation daily? • Calculate the standard deviation of this distribution.

  11. Solution X=number of workers take public transportation daily.

  12. Extra Exercise • In a large shipment of automobile tires, 5% have a certain flaw. A sample of four tires was chosen to be installed on a car. Let X be the random variable of tires have flaw. • What is probability that at least one of the tires has a flaw? • What is the probability that exactly three of the tires have no flaw?

  13. The Poisson Distribution • A random variable X has a Poisson distribution and it is referred to as a Poisson random variable if and only if its probability distribution is given by

  14. λ (Greek lambda) is the long run mean number of events for the specific time or space dimension of interest. • A random variable X having a Poisson distribution can also be written as

  15. REMEMBER!!!!!!!!!!!!!!!!!!!! Average Rate Mean

  16. Example 2 Given that , find

  17. Try to find the probabilities using table. Do you get the same answer??? solutions

  18. Example 3 Suppose that the number of errors in a piece of software has a Poisson distribution with parameter . Find a) the probability that a piece of software has no errors. b) the probability that there are three or more errors in piece of software . c) the mean and variance in the number of errors.

  19. solutions

  20. Exercise 2 • Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways • Find the probability of receiving three calls in a 5-minutes interval time. • Find the probability of receiving more than two calls in 15 minutes.

  21. Exercise 3 • An average of 15 aircraft accidents occurs each year. Find • The mean, variance and standard deviation of aircraft accident per month. • The probability of no accident during a months.

  22. IMPORTANT!!!!

  23. NORMAL DISTRIBUTION

  24. INTRODUCTION

  25. Normal Distribution

  26. The Normal Distribution has: • mean = median = mode • symmetry about the center • 50% of values less than the mean and 50% greater than the mean

  27. Applications of normal distribution • Many naturally occurring random processes tend to have a distribution that is approximately normal.  Examples can be found in any field, these include: • heights and weights of adults • length and width of leaves of the same species • actual weights of rice in 5 kg bags sold in supermarkets

  28. The Standard Normal Distribution • The normal distribution with parameters and is called a standard normal distribution. • A random variable that has a standard normal distribution is called a standard normal random variable and will be denoted by .

  29. Standardizing A Normal Distribution

  30. Why Standardize ... ? • It can help you make decisions about your data. • It also makes life easier because we only need one table (the Standard Normal Distribution Table), rather than doing calculations individually for each value of mean and standard deviation.

  31. Example: Professor Willoughby is marking a test. • Here are the students results (out of 60 points): 20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17 • Most students didn't even get 30 out of 60, and most will fail.The test must have been really hard, so the Prof decides to Standardize all the scores and only fail people 1 standard deviation below the mean. • The Mean is 23, and the Standard Deviation is 6.6, and these are the Standard Scores: • -0.45, -1.21, 0.45, 1.36, -0.76, 0.76, 1.82, -1.36, 0.45, -0.15, -0.91 • Only 2 students will fail (the ones who scored 15 and 14 on the test)

  32. Standard normal distribution Total area =1

  33. Example 4 Determine the probability or area for the portions of the Normal distribution described. (using the normal table)

  34. solutions • Using table

  35. Calculate the probabilities using calculator • Mode: SD • 1 • Shift 3 • P(1) for • Q (2) for • R (3) for

  36. Example 4 Determine the probability or area for the portions of the Normal distribution described. (using the calculator)

  37. Example 5 The masses of a well known brand of breakfast cereal are normally distributed with mean of 250g and standard deviation of 4g. Find the probability of a packet containing more than 254.4g.

  38. solutions • Let X be the r.v. “masses of cereal in grams” where X~N(250, 16).

  39. EXERCISE 3 A battery has a lifetimes which are normally distributed with a mean of 62 hours and a standard deviation of 3 hours. What is the probability of battery lasting less than 68 hours?

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