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Chapter 5

Chapter 5. Intro to random variables and probability distributions. Statistical experiment of observation is any process by which measurements are obtained. Measuring daily rainfall in inches Counting the number of defective light bulbs in a case of bulbs.

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Chapter 5

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  1. Chapter 5

  2. Intro to random variables and probability distributions • Statistical experiment of observation is any process by which measurements are obtained. • Measuring daily rainfall in inches • Counting the number of defective light bulbs in a case of bulbs.

  3. Let’s use the example of the defective light bulbs: x = # of defective light bulbs, this is an example of a quantitative variable. x is a random variable because the value that x takes on in a given is a chance or random outcome.

  4. Two types of random variables • Discrete – observations of a quantitative random variable can take on only a finite # of values or a countable number of values. • Continuous – observations of a quantitative random variable can take on any of the countless values in a line interval.

  5. Discrete or continuous random variables? • Measure the height of a student selected at random. • Continuous • Count the number of students in AP classes at a school selected at random. • Discrete

  6. All random variables have a probability distribution • Discrete • The probability distribution of a discrete random variable has a probability assigned to each value. • The sum of these probabilities is 1.

  7. Discrete probability distribution One of the elementary tools of cryptanalysis is to use relative frequencies of occurrence of different letters in the alphabet to break standard English alphabet codes. Large samples of plain text such as newspaper stories generally yield about the same relative frequencies for letters. A sample 1000 letters long yielded the information in the table:

  8. Letter Freq. Letter Freq. A 73 N 78 B 9 O 74 C 30 P 27 D 44 Q 3 E 130 R 77 F 28 S 63 G 16 T 93 H 35 U 27 I 74 V 13

  9. Use the relative frequencies to compute the probabilities of the vowels. P(A) = 0.073 P(E) = 0.130 P(I) = 0.074 P(O) = 0.074 P (U) = 0.027 If a letter is selected at random from a newspaper story, what is the probability that the letter will be a vowel? P(a,e,i,o,u) = 0.378

  10. Mean and standard deviation of a discrete probability distribution µ=  (x(P(x)) -expected value =   ( x - )2P(x) x = value of the random variable P(x) = probability of that value  = sum

  11. Are we influenced to buy a product by an ad we saw on TV? The National Infomercial Marketing Association determined the number of times buyers of a product watched a TV infomercial before purchasing the product. The table below shows the results.

  12. Because the events are mutually exclusive and all the percentages add up to 100%; we can treat the information as an approximation of the probability distribution. Compute the mean and standard deviation of the distribution.

  13.  =  (x(P(x)) = 0.27+0.62+0.54+0.36+0.54+0.36+0.75=2.54 =   ( x - )2P(x) = 1.37113748

  14. Linear functions of a random variable Let x be a random variable with mean  and standard deviation . Then L = a + bx has a mean, variance, and standard deviation as follows. L = a + b  L2 = b22 L =  b22 = |b| 

  15. Linear combinations of independent random variables Let x1 and x2 be independent random variables with respective means 1 and 2 and variances 12 and 22 W = ax1 + bx2  w = a 1 + b 2 w2 = a212 + b222 w = a212 + b222

  16. Let x1 and x2 be independent random variables with respective means 1 = 75 and 2 =50and standard deviations 1= 16 and 2= 9. a.) Let L = 3 + 2x Find L L2 L b.) Let W = 3x1 - 2x2 Find L L2 L

  17. 5.2 Binomial Probabilities Features of a binomial experiment • Fixed # of trials, n. • The n trials are independent and repeated under identical conditions • Each trial has two outcomes (Success s and Failure f)

  18. 4. The probability of success is the same. P(success) = p P(failure) = q p + q = 1 q = 1 – p 5. Used to find the probability of r successes out of n trials.

  19. According to the Textbook of Medical Physiology, 5th Edition, by Arthur Guyton, 9% of the population has blood type B. Suppose we choose 18 people at random from the population and the blood type of each. What is the probability that 3 of these people have blood type b? Find p, q, n, and r.

  20. In this experiment, we are observing whether of not a person has type B blood. A success in this case would be that the person has type B blood. A failure would be if the person does not have type B blood. P(success) = 0.09 P(failure) = 0.91 n = 18 r = 3

  21. Computing probabilities for a binomial experiment n = 3 p =.25 q = .75 Outcome | probability of outcome | r

  22. General formula for binomial probability distribution P(r) = Cn,r pr qn – r Where Cn,r is the binomial coefficient.

  23. Privacy is a concern for many users of the Internet. One survey showed that 59% of Internet users are somewhat concerned about the confidentiality of their e-mail. Based on this information, what is the probability that for a random sample of 10 Internet users, 6 are concerned about the privacy of their e-mail? n = p = q = r =

  24. n = p = q = r = n = 10 p = 0.59 q = 0.41 r = 6 P(6) = 0.25

  25. A biologist is studying a new hybird tomato. It is known that the seeds of this hybird have probability of 0.70 of germinating. The biologist plant 10 seeds, what is the probability that exactly 8 seeds will germinate? What is the probability that at least 8 seeds will germinate? Use table 3

  26. Expression Inequality # or more successes At less # successes r > # No fewer than # successes Not less than # successes _____________________________________ # or fewer successes At most # successes r < # No more than # successes The number of successes does not exceed#

  27. More than # successes r > # The number of successes exceeds# _____________________________________ Fewer than # successes r < # The number of successes is not as large as #

  28. Calculator Binompdf (n,p,r) - for single trial Binomcdf (n,p,r) – for cumulative probability that are r or fewer successes.

  29. 5.3 Additional Properties of the Binomial Distribution • Binomial distribution-tells the probability of r successes out of n trials. A waiter has learned from experience that the probability that a lone diner will leave a tip is only 0.7. During 1 lunch hour he serves 6 people who are dining by themselves. Make a graph of the binomial probability distribution that shows the probabilities that 0,1,2,3,4,5 or all 6 diners leave tips.

  30. r | P(r) 0 0.001 1 0.010 2 0.060 3 0.185 4 0.324 5 0.303 6 0.118 The height of the bar over a particular r value tells the probability of that r. (the area of the bar also tells the probability)

  31. For the binomial distribution μ = np - expected # of successes r σ = √npq - is the standard deviation for the number of successes r. Where n is the number of trials p is the probability of successes q is the probability of failure

  32. Compute the mean and standard deviation for the distribution of the 6 lone diners leaving tips. n = 6 p = 0.7 q = 0.3 μ = np = 4.2 σ = √ npq = 1.1224

  33. 5.4 The Geometric and Poisson Probability Distribution • Used if we have an experiment where we repeat binomial trials until we get our first success and stop. • If n is the # of trial on which we have the 1st success • n could be 1, 2, 3, 4, …

  34. What is the probability our 1st success comes on the nth trial? • The answer is given by the Geometric probability distribution P(n) = p (1 – p)n-1 n = # of the trial on which the 1st success occurs p = the probability of success on each trial (p must be the same for each trial.)

  35. Mean for the Geometric Distribution μ = 1/p Standard deviation for the Geometric Distribution σ = √(1 – p) / p

  36. An automobile assembly plant produces sheet metal décor panels. As the panel passes a robot, a mechanical arm will perform spot welding at different locations. Each location has a magnetic dot painted where the weld is to be made. Experience shows that on each trial the robot is 85% successful at locating the dot. If it cannot locate the magnetic dot, it is programmed to try again.

  37. The robot will keep trying until it finds the dot of the door panel passes out of the robots reach. What is the probability that the robot’s first success will be on attempts n = 1, 2, or 3? P(1) = .85 P(2) = 0.1275 P(3) = 0.0191

  38. The assembly line moves so fast that the robot has a maximum of only three chances before the door panel is out of reach. What is the probability that the robot will be successful before the door is out of reach? P(1or2or3) = P(1) +P(2)+ P(3) = .85 +.1275 +.191 = .9966

  39. Probability that the robot will not be able to locate the correct spot within 3 tries? 1 – 0.9966 = 0.0034 If 10,000 panels are made, what is the expected number of defective panels? (10,000)(0.0034) = 34 defective panels

  40. Poisson Probability Distribution • n gets larger, p gets smaller Poisson was French mathematician He studied probabilities of rare events (space, time, volume) Poisson Distribution applies to accident rates, arrival times, defect rates.

  41. As with the binomial we assume only two outcomes. (occurs, does not occur) Events are independent, so that one success does not change the probability of another success.

  42. Poisson Distribution Let λ = the mean number of successes over time, volume, area,… Let r = the # of successes in a corresponding interval of time, volume, area,… P(r) = probability of r successes in the interval of time, volume, area,…

  43. P(r) = e-λλr r! (note) e = 2.7183 μ=λ σ = √λ

  44. Pyramid lake is located in Nevada on the Paiute Indian Reservation. The lake contains some of the world’s largest cutthroat trout. 8 to 10 lbs not uncommon 12 to 15 lbs trophies Fish biologists published that the Nov. catch for boat fisherman - total fish per hour = 0.667

  45. Suppose you decide to fish Pyramid lake for 7 hours during the month of Nov. A.) use the information provided by the biologist to find the probability distribution for r the # of fish (all sizes) you catch in a period of 7 hours. λ = o.667/1 adjust for 7 hours λ = 4.669/7 = 4.7 for 7hrs period

  46. P(r) = e-4.7(4.7)rr! B.) What is the probability that in 7 hours you will catch 0, 1, 2, or 3 fish of any size? P(0) = 0.0091 P(1) = 0.0427 P(2) = 0.1005 P(3) = 0.1574

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