Topic 6: Probability Distribution

1. Topic 6:Probability Distribution Objectives: Know the characteristics for each of the distribution. Be able to calculate the probability either by using binomial, Poisson and normal distribution. Be able to find mean and variance of each distribution. Be able to use Binomial (Table 1), Poisson (Table 2) and Normal (Table 4) distribution table to determine the probability.

2. Binomial Distribution The Four Conditions of a Bernoulli Process There are n identical trials. Each trial has only two possible outcomes. The probabilities of the two outcomes remain constant. The trials are independent.

3. What is it�? Consider the experiment consisting of 10 tosses of a coin. Determine whether or not it is a binomial experiment. There are a total of 10 trials (tosses), and they are all identical. All 10 tosses are performed under identical condition. Each trial (toss) has only two possible outcomes: a head (success) and a tail (failure). P(head) = � and P(tail) = � remain the same for each toss. The sum of these two probabilities is 1.0. The trials (tosses) are independent. The outcome of one trial does not affect the outcome of another trial.

4. Another example� Five percent of all VCRs manufactured by a large electronics company are defective. Three VCRs are randomly selected from the production line of this company. The VCRs are inspected to determine whether each of them is defective or good. Is this experiment a binomial experiment

5. Formula The probability distribution of the binomial random variable X obtaining r successes in a Bernoulli process consisting of n trials with success probability p and failure probability q = 1 - p is: for r = 0, 1, 2,�, n where: n = total number of trials p = probability of success q = 1 - p = probability of failure r = number of successes in n trials n - r = number of failures in n trials

6. Formula In any Binomial experiment with n Bernoulli trials, The mean and standard deviation for a binomial distribution are: ? = np and

7. Still blur�? Let say, Five percent of all calculators manufactured by a large electronics company are defective. A quality control inspector randomly selects three calculators from the production line. What is the probability that exactly one of these calculators is defective?

8. Example Example 1: For Bernoulli trials compute the following probabilities: 2 successes in 4 trials with p = 0.3. successes in 5 trials with p = 1/3. 4 failures in 6 trials with p = 0.2.

9. Using The Table Of Binomial Probabilities

10. Example Example 2: Suppose that 20% of all copies of a particular textbook fail a certain binding strength test. Let X denote the number among 15 randomly selected copies that fail test. Find the probability that: at most 8 fail the test, exactly 8 fail the test, at least 8 fail the test, between 4 to 8 fail the test.

11. Another one� Example 3: A traffic control engineer reports that 75% of the vehicles passing through a checkpoint are from within the state. Of the next 9 vehicles, find the probability that: from 3 to 6 fewer than 4 more than 5 between 2 and 6 are from within the state? Find the mean and variance.

12. Exercise According to a study, 25% of CEO said that their favorite luxury car is Mercedes. Assume that this result holds true for the current population of all Malaysia CEOs. A sample of 40 CEOs is selected. Let x denote the number of CEOs in this sample who hold this view. Find the mean and standard deviation of the probability distribution of x

13. Poisson Distribution Three Conditions of a Poisson Probability Distribution x is a discrete random variable. The occurrences are random (do not follow any pattern). The occurrences are independent (consider with respect to an interval).

14. What is it�? Example of events which might follow Poisson distribution are: The number of accidents on a particular stretch of road in one day. The number of telephone calls made in a given minute. The number of typing errors in one page

15. Formula For a Poisson probability distribution, the probability of x occurrences in an interval is: x = 0, 1, 2, � where ? (lambda) = mean number of occurrences in that interval and the value of e ? 2.71828.

16. More formula� The mean and variance of the Poisson distribution poi(x; ?t), both have the value ?t.

17. Example Example 4: During a laboratory experiment the average no of radioactive particles passing through a counter in 1 millisecond is 4. What is the probability that: 6 particles enter the counter in a given millisecond at most 1 particle enter the counter in a given millisecond? at least 2 particles enter the counter in 2 millisecond?

18. Another exercise� On average, a household receives 9.5 telemarketing phone calls per week. Using the Poisson distribution formula, find the probability that a randomly selected household receives exactly six telemarketing phone calls during a given week.

19. Using The Table of Poisson Distribution Function

20. Example Example 5: On average a certain intersection results in 3 traffic accidents per month. What is the probability that for any given month at this intersection exactly 5 accidents will occur? less than 3 accidents will occur? at least 2 accidents will occur? Find the mean and variance.

21. Normal Distribution Properties of a normal probabilities distribution The total area under the curve is 1.0. The curve is symmetric about the mean ?. The two tails of the curve extend indefinitely.

22. The graphs�

23. What else�? The Standard Normal Distribution For the standard normal distribution, the value of the mean in equal to zero, and the value of standard deviation is equal to 1. Formula to find z: where ? = mean and ? = standard deviation

24. Using Table Of Normal Probabilities The probabilities can be treated as areas and are obtained by using the table of the standard normal distribution. The z score is calculated and entered into the table to obtain required areas or probability under the curve less than or equal to the number represented by the z score.

25. Example Example 6: Given a standard normal distribution, find the area under the curve that lies to the left of z = 1.84. to the right of z = 2.31. greater than z = �0.65 less than z = -2.34. between z = 1.97 and z = 0.86. between z = -1.13 and z = 0.89. between z = -3.20 and z = -0.22 greater than z = 5.67 between z = �5.0 and z = 0.

26. Converting An X Value To A Z-Value For a normal random variable x, a particular value of x can be converted to its corresponding z value by using the formula: where ? and ? are the mean and standard deviation of the normal distribution of x.

27. Example Example 7: Given a normal distribution with mean = 50 and standard deviation = 10, find the probability that X a value: between 45 and 62. less than 55

28. Another scenario Determining The Z And X Values When An Area Under The Normal Distribution Curve Is Known Convert z value to x value by using the formula:

29. Example Example 8: Find the value of k such that P(Z > k) = 0.3015. P(-k < Z < -0.18) = 0.4197. P(Z < k) = 0.0427 P(|Z| < k ) = 0.99

30. Application Example Example 9: Finding a source of supply for factory workers is especially difficult when a nation�s unemployment rate is relatively low. Under these conditions, available labor is scarce, and wages also tend to increase. It is known that, in April 1998 the average production worker in Melaka earned $13.47 an hour, up 3.1% from a year earlier. That percentage increase was more than all of 1997 and 1996 combined. Suppose production worker�s wages are normally distributed with a standard deviation of $4.75. What percentage of the nation�s production workers earn: between $11.00 and $15.00 an hour? between $8.00 and $19.00 an hour? over $20.00 an hour? less than $6.00 an hour?

31. Last one� Example 10: A soft-drink machine is regulated so that it discharges an average of 200 ml per cup. If the amount of drink is normally distributed with a standard deviation equal to 15 ml, What fraction of the cups will contain more than 224 ml? What is the probability that a cup contain between 191 and 209 ml? How many cups will probably overflow if 230ml cups are used for the next 1000 drinks? Below what value do we get the smallest 25% of the drinks?