1.  A probability function - function when probability values are assigned to all possible numerical values of a random variable (X).  Individual probability values may be denoted by the symbol P(X=x), in the discrete case, which indicates that the random variable can have various specific values.  All the probabilities must be between 0 and 1; 0≤ P(X=x)≤ 1  The sum of the probabilities of the outcomes must be 1. ∑ P(X=x)=1  It may also be denoted by the symbol f(x), in the continuous, which indicates that a mathematical function is involved. Probability Distributions

2. Continuous Probability Distributions Binomial Poisson Probability Distributions Discrete Probability Distributions Normal

3. B INOMIAL D ISTRIBUTION An experiment in which satisfied the following characteristic is called a binomial experiment: The random experiment consists of n identical trials. Each trial can result in one of two outcomes, which we denote by success, S or failure, F. The trials are independent. 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. Example: No. of getting a head in tossing a coin 10 times. o Binomial distribution is written as X ~ B(n,p),

4. A binomial experiment consist of n identical trial with probability of success, p in each trial. The probability of success in n trials is given by Note: The Mean and Variance of X If X ~ B(n, p), then Mean : Variance : Std Deviation : where n is the total number of trials, p is the probability of success and q is the probability of failure.

5. E XAMPLE :

6. S OLUTION :

7. C UMULATIVE B INOMIAL DISTRIBUTION When the sample is relatively large, tables of Binomial are often used. Since the probabilities provided in the tables are in the cumulative form the following guidelines can be used: Bin. table

8. E XAMPLE : Bin. table

9. EXAMPLE: The probability that boards purchased by a cabinet manufacturer are unusable for building cabinets is 0.10. The cabinet manufacturer bought eleven boards, what is the probability that a) Four or more of the eleven boards are unusable for building cabinets? b) At most two of the eleven boards are unusable for building cabinets? c) None of the eleven boards are unusable for building cabinets? Solution: X : The number of unusable boards X ~ B(11, 0.10) a) b) c) Bin. table

10. Exercise: 1. Given that X~B(20, 0.25) using tables of binomial probabilites, find 2. According to a survey, 45% of college students wear contact lense a) What is the probability that exactly 3 of a random sample of 5 college student wear contact lenses? b) What is the probability that at least 7of a random sample of 15 college student wear contact lenses? c) What is the probability that not more than 7 of a random sample of 15 college student wear contact lenses? Bin. table

11. T HE P OISSON D ISTRIBUTION  Poisson distribution is the probability of a given number of events occurring in a fixed interval of time / space and other specified intervals such as distance, area /measurement or volume. Examples: 1. The number of cars passing a toll booth in one hour (time). 2. The number of defects in a square meter of fabric (area).

12.  A random variable X has a Poisson distribution and it is written as with  Probability distribution function of Poisson is given by

13. E XAMPLE : Given that, find

14. S OLUTION :

15. E XAMPLE : Poi. table

16. Exercise: 1. Given that, using tables of Poisson probabilities, find 2. The numbers of cars sold by a new car dealer follows a Poisson distribution with a mean of 13.5 cars sold in three days. a) What is the probability that at least 6 cars a sold today? b) Find the mean and standard deviation of Y, the number of cars sold in two days. What is the probability that fewer than 10 cars sold in two days? c) Find the mean and standard deviation of W, the number of cars sold in four days. What is the probability that at most 18 cars are sold in 4 days? Poi. table

17. P OISSON A PPROXIMATION OF B INOMIAL P ROBABILITIES Example: Given that X~B (1000, 0.004). Find a) P (X=7) b) P (X<9) Solution: Since n = 1000 >30, np = 1000(0.004) = 4 < 5, thus Poisson approximation is used with. Therefore, a) b) Applicable to used Poisson distribution when the Binomial experiment / trial has the following : Poi. table

18. Exercise: 1. Given that, find 2. 0.4% of all 9 city’s voters are not in favour of a certain candidate for mayor. Suppose a poll of 1000 voters in the city is taken. Find the probability that 10 or more voters do not favour this candidates. Poi. table

19. T HE N ORMAL D ISTRIBUTION Numerous continuous random variables have distribution closely resemble the normal distribution. The normal distribution can be used to approximate various discrete prob. dist.

20. T HE N ORMAL D ISTRIBUTION ‘Bell Shaped’ Symmetric about the mean Mean, Median and Mode are Equal - Location is determined by the mean, μ - Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   Mean = Median = Mode X f(X) μ σ

21. The Standard Normal Distribution Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (Z) Need to transform X units into Z units using The standardized normal distribution (Z) has a mean of 0, and a standard deviation of 1,. Z is denoted by Thus, its density function becomes

22. P ATTERNS FOR F INDING A REAS UNDER THE S TANDARD N ORMAL C URVE

23. E XAMPLE : Z table

24. E XERCISES : Determine the probability or area for the portions of the Normal distribution described. Z table

25. S OLUTIONS :

26. E XAMPLE : Z table

27. E XERCISES : Z table

28. S OLUTIONS :

29. E XAMPLE : Suppose X is a normal distribution N(25,25). Find Solutions: Z table

30. E XERCISES : Z table

31. E XAMPLE : Z table to

32. Exercise: At a certain community college, the time that is required by students to complete the math competency examination is normally distributed with a mean of 57.6 minutes and a standard deviation of 8 minutes. Find the probability that a student takes a) Longer than 1 hour to complete the examination b) Between 56 minutes and I hour to complete the examination Z table

33. N ORMAL A PPROXIMATION OF THE B INOMIAL D ISTRIBUTION  When the number of observations or trials n in a binomial experiment is relatively large, the normal probability distribution can be used to approximate binomial probabilities. A convenient rule is that such approximation is acceptable when

34. C ONTINUOUS C ORRECTION F ACTOR  The continuous correction factor needs to be made when a continuous curve is being used to approximate discrete probability distributions.  0.5 is added or subtracted as a continuous correction factor according to the form of the probability statement as follows :

35. Example: In a certain country, 45% of registered voters are male. If 300 registered voters from that country are selected at random, find the probability that at least 155 are males. Solutions : Z table

36. E XERCISES : Suppose that 5% of the population over 70 years old has disease Suppose a random sample of 9600 people over 70 is taken. What is the probability that fewer than 500 of them have disease A? Answer: Z table

37. N ORMAL A PPROXIMATION OF THE P OISSON D ISTRIBUTION  When the mean of a Poisson distribution is relatively large, the normal probability distribution can be used to approximate Poisson probabilities. A convenient rule is that such approximation is acceptable when

38. E XAMPLE : A grocery store has an ATM machine inside. An average of 5 customers per hour comes to use the machine. What is the probability that more than 30 customers come to use the machine between 8.00 am and 5.00 pm? Solution: Z table

39. Exercise: An average of 10 patients are admitted per day to the emergency room of a big hospital. What is the probability that less 75 than patients are admitted to the emergency room in 7 days? Z table

40. Continue Lecture 3