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### Chapter6_ContinuousProbabilityDistributions

Course: BIT 2405, Spring 2008
School: Virginia Tech
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6 Chapter Continuous Probability Distributions Continuous Probability Distribution Probability Distribution for r.v. X is...? Continuous random variable? Continuous Probability Distribution (continued) Requirement with probability dist? If X is continuous, then... Continuous Probability Distribution (continued) Consider the interval (a, b) where: - ab If a = b, then? The interval is a single point and...

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6 Chapter Continuous Probability Distributions Continuous Probability Distribution Probability Distribution for r.v. X is...? Continuous random variable? Continuous Probability Distribution (continued) Requirement with probability dist? If X is continuous, then... Continuous Probability Distribution (continued) Consider the interval (a, b) where: - ab If a = b, then? The interval is a single point and P(single point)=0. We want prob. that X (a, b) P(a < X < b)... which is the area under the curve over the interval a to b. Normal Distribution aka Gaussian distribution Most useful and most commonly used Plays an important role in statistics b/c: Numerous phenomena seem to follow the normal dist. or can be approx. by it. Can be used to approx. discrete prob. dist. Plays very important role in classical statistical inference Normal Distribution (continued) Properties Normal curve is bell-shaped Mean Median Mode Perfectly symmetric about the mean (central value) Normal Distribution (continued) Characterized by: , the population mean , the population standard deviation Normal probability density function: 1 x 2 2 f (x) 1 e 2 , for all x Normal Distribution (continued) Normal Distribution (continued) Function allows us to determine the probability that x falls in a given interval through numerical integration b b f ( x )dx a a 1 e 2 1 x 2 2 dx , for all x Normal Distribution (continued) Normal Distribution (continued) Integration method difficult & time-consuming Pre-generated table would be appropriate Unfortunately, there are an infinite number of normal tables since there are an infinite number of combinations for and . Fortunately, ALL normal probability distributions can be represented in ONE table. Each value of X can be identified by the number of standard deviations it is away from the mean Normal Distribution & Empirical Rule Standard Normal Distribution Standard normal distribution (Z) has: mean ( ) equal to 0 standard deviation ( ) equal to 1 We write: Z ~ N(0, 1) which is read as "Z is normally distributed with = 0 and = 1. Standard Normal Distribution (continued) All normal probability distributions any X~N( , ) can be converted to a standard normal distribution Z~N(0,1) by standardization. If X~N( , ), X can be standardized to a z-score where Z~N(0,1) by z X Standard Normal Distribution (continued) Standard Normal Curve Gives cumulative probabilities P( Z < z) To use the table: Read the value of z down the leftmost column and across the top row Find the corresponding value of the probability in the body of the table Cumulative Probabilities for the Standard Normal Distribution Cumulative probability z 0 z -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.00 0.0013 0.0019 0.0026 0.0035 0.0047 0.0062 0.0082 0.0107 0.0139 0.0179 0.0228 0.0287 0.0359 0.0446 0.0548 0.0668 0.0808 0.0968 0.1151 0.1357 0.1587 0.1841 0.2119 0.2420 0.2743 0.3085 0.3446 0.3821 0.4207 0.4602 0.5000 0.01 0.0013 0.0018 0.0025 0.0034 0.0045 0.0060 0.0080 0.0104 0.0136 0.0174 0.0222 0.0281 0.0351 0.0436 0.0537 0.0655 0.0793 0.0951 0.1131 0.1335 0.1562 0.1814 0.2090 0.2389 0.2709 0.3050 0.3409 0.3783 0.4168 0.4562 0.4960 0.02 0.0013 0.0018 0.0024 0.0033 0.0044 0.0059 0.0078 0.0102 0.0132 0.0170 0.0217 0.0274 0.0344 0.0427 0.0526 0.0643 0.0778 0.0934 0.1112 0.1314 0.1539 0.1788 0.2061 0.2358 0.2676 0.3015 0.3372 0.3745 0.4129 0.4522 0.4920 0.03 0.0012 0.0017 0.0023 0.0032 0.0043 0.0057 0.0075 0.0099 0.0129 0.0166 0.0212 0.0268 0.0336 0.0418 0.0516 0.0630 0.0764 0.0918 0.1093 0.1292 0.1515 0.1762 0.2033 0.2327 0.2643 0.2981 0.3336 0.3707 0.4090 0.4483 0.4880 0.04 0.0012 0.0016 0.0023 0.0031 0.0041 0.0055 0.0073 0.0096 0.0125 0.0162 0.0207 0.0262 0.0329 0.0409 0.0505 0.0618 0.0749 0.0901 0.1075 0.1271 0.1492 0.1736 0.2005 0.2296 0.2611 0.2946 0.3300 0.3669 0.4052 0.4443 0.4840 0.05 0.0011 0.0016 0.0022 0.0030 0.0040 0.0054 0.0071 0.0094 0.0122 0.0158 0.0202 0.0256 0.0322 0.0401 0.0495 0.0606 0.0735 0.0885 0.1056 0.1251 0.1469 0.1711 0.1977 0.2266 0.2578 0.2912 0.3264 0.3632 0.4013 0.4404 0.4801 0.06 0.0011 0.0015 0.0021 0.0029 0.0039 0.0052 0.0069 0.0091 0.0119 0.0154 0.0197 0.0250 0.0314 0.0392 0.0485 0.0594 0.0721 0.0869 0.1038 0.1230 0.1446 0.1685 0.1949 0.2236 0.2546 0.2877 0.3228 0.3594 0.3974 0.4364 0.4761 0.07 0.0011 0.0015 0.0021 0.0028 0.0038 0.0051 0.0068 0.0089 0.0116 0.0150 0.0192 0.0244 0.0307 0.0384 0.0475 0.0582 0.0708 0.0853 0.1020 0.1210 0.1423 0.1660 0.1922 0.2206 0.2514 0.2843 0.3192 0.3557 0.3936 0.4325 0.4721 0.08 0.0010 0.0014 0.0020 0.0027 0.0037 0.0049 0.0066 0.0087 0.0113 0.0146 0.0188 0.0239 0.0301 0.0375 0.0465 0.0571 0.0694 0.0838 0.1003 0.1190 0.1401 0.1635 0.1894 0.2177 0.2483 0.2810 0.3156 0.3520 0.3897 0.4286 0.4681 0.09 0.0010 0.0014 0.0019 0.0026 0.0036 0.0048 0.0064 0.0084 0.0110 0.0143 0.0183 0.0233 0.0294 0.0367 0.0455 0.0559 0.0681 0.0823 0.0985 0.1170 0.1379 0.1611 0.1867 0.2148 0.2451 0.2776 0.3121 0.3483 0.3859 0.4247 0.4641 Cumulative Probabilities for the Standard Normal Distribution z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 0.00 0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772 0.9821 0.9861 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.9981 0.9987 0.01 0.5040 0.5438 0.5832 0.6217 0.6591 0.6950 0.7291 0.7611 0.7910 0.8186 0.8438 0.8665 0.8869 0.9049 0.9207 0.9345 0.9463 0.9564 0.9649 0.9719 0.9778 0.9826 0.9864 0.9896 0.9920 0.9940 0.9955 0.9966 0.9975 0.9982 0.9987 0.02 0.5080 0.5478 0.5871 0.6255 0.6628 0.6985 0.7324 0.7642 0.7939 0.8212 0.8461 0.8686 0.8888 0.9066 0.9222 0.9357 0.9474 0.9573 0.9656 0.9726 0.9783 0.9830 0.9868 0.9898 0.9922 0.9941 0.9956 0.9967 0.9976 0.9982 0.9987 0.03 0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967 0.8238 0.8485 0.8708 0.8907 0.9082 0.9236 0.9370 0.9484 0.9582 0.9664 0.9732 0.9788 0.9834 0.9871 0.9901 0.9925 0.9943 0.9957 0.9968 0.9977 0.9983 0.9988 0.04 0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.7389 0.7704 0.7995 0.8264 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382 0.9495 0.9591 0.9671 0.9738 0.9793 0.9838 0.9875 0.9904 0.9927 0.9945 0.9959 0.9969 0.9977 0.9984 0.9988 0.05 0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023 0.8289 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 0.9505 0.9599 0.9678 0.9744 0.9798 0.9842 0.9878 0.9906 0.9929 0.9946 0.9960 0.9970 0.9978 0.9984 0.9989 0.06 0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.7764 0.8051 0.8315 0.8554 0.8770 0.8962 0.9131 0.9279 0.9406 0.9515 0.9608 0.9686 0.9750 0.9803 0.9846 0.9881 0.9909 0.9931 0.9948 0.9961 0.9971 0.9979 0.9985 0.9989 0.07 0.5279 0.5675 0.6064 0.6443 0.6808 0.7157 0.7486 0.7794 0.8078 0.8340 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.9525 0.9616 0.9693 0.9756 0.9808 0.9850 0.9884 0.9911 0.9932 0.9949 0.9962 0.9972 0.9979 0.9985 0.9989 0.08 0.5319 0.5714 0.6103 0.6480 0.6844 0.7190 0.7517 0.7823 0.8106 0.8365 0.8599 0.8810 0.8997 0.9162 0.9306 0.9429 0.9535 0.9625 0.9699 0.9761 0.9812 0.9854 0.9887 0.9913 0.9934 0.9951 0.9963 0.9973 0.9980 0.9986 0.9990 0.09 0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986 0.9990 Standard Normal Table Examples P(0 Z 1) P(Z 2.37) P(Z > -0.45) P(-1 Z 2.15) P(-1.6 Z 1.6) P(2 Z 3.5) Standard Normal Table Examples Standard Normal Table Examples Standard Normal Table Examples Normal Distribution Word Poblems 1. Make sure X~N Identify and 2. Write out the prob. to be determined (pictures) 3. Standardize X 4. Use Std. Normal Table to find probability Normal Distribution Word Problems Example 1 Individual weights of a variety of hand wrapped cigars are bell-shaped with a mean of 24 grams a and variance of 9 grams squared. If you randomly selected a cigar, what is the probability that the cigar weighs between 15.25 grams and 19.5 grams? Normal Distribution Word Problems Example 1 Normal Distribution Word Problems Example 2 A toy manufacturer has found that the number of hours needed for an assembly line to produce 1000 toy products each day is approximately normal, with a mean of 4 hours and a variance of 4 hours squared. What is the probability that the 1000 toys will be produced in no more than 4.5 hours? Normal Distribution Word Problems Example 2 Normal Distribution Word Problems Example 3 The grade point averages of a large population of college students are approximately normally distributed with a mean of 2.4 and a standard deviation of 0.8. What is the probability that a randomly selected student will have a grade point average in excess of 3.0? Normal Distribution Word Problems Example 3 Normal Distribution Word Problems Example 4 A machine used to regulate the amount of dye dispensed for mixing shades of paint can be set so that it discharges an average of milliliters of dye per can of paint. The amount of dye discharged is known to have a normal distribution with variance equal to 0.160. If more than 6 milliliters of dye are discharged when making a particular shade of blue paint, the shade is unacceptable. Determine the setting of so that no more than 1% of the cans of paint will be unacceptable. Normal Distribution Word Problems Example 4 Normal Distribution Fractiles Given a probability and want to find the value(s) of the normal random variable that corresponds to that probability Example: Average daily production of a manufacturing plant is known to be normally distributed, and the manager wants to pay an incentive bonus to the crew when their level of production exceeds 85% of the daily production. Normal Distribution Fractiles (continued) We are not interested in the probability of an event We want to calculate the production level This type of problem is called a fractile. To solve, we just work backwards. Normal Distribution Fractiles (continued) With the example... Suppose, you are given a normal pop. with =100 and =10. Find the value of X such that 85% of the X's are less than or equal to this value. Normal Distribution Fractiles (continued) Normal Distribution Fractiles (continued) Locate prob. in middle of table Find corresponding z-score (use closest prob. in table) Un-standardize Normal Distribution Steps to Working Fractiles 1. 2. 3. 4. 5. 6. 7. 8. Identify a normal dist, and Identify the probability (picture) Determine the fractile Determine the prob. table value Look up prob. table value Find corresponding value of z Decide if z is + or Unstandardize z to get X Normal Distribution Fractile Example 1 The length of time required to complete a college achievement test is found to be normally distributed with a mean of 70 minutes and a standard deviation of 12 minutes. When should the test be terminated if we wish to allow sufficient time for 90% of the students to complete the test? Normal Distribution Fractile Example 1 Normal Distribution Fractile Example 2 A manufacturing plant utilizes 3000 electric light bulbs that have a length of life that is normally distributed with a mean of 500 hours and a standard deviation of 50 hours. To minimize the number of bulbs that burn out during operation hours, all the bulbs are replaced after a given period of operation. How often should the bulbs be replaced if we wish not more than 1% of the bulbs to burn out between replacement periods? Normal Distribution Fractile Example 2 Normal Approximation to Binomial Binomial Distibution Sample of n discrete trials One of two possible outcomes (S/F) P(S) = p X, the random variable, # of "successes" in n trials P(X = x) using binomial formula P( X x) n x p (1 p) n x x Normal Approximation to Binomial (continued) If n is large, prob. are difficult to calculate Often times, we want P(X xi), P(Xxi), or P(xiXxj). Normal approximation to binomial is best for large sample (large n) ROT: Normal distribution should only be used to approximate the binomial distribution if n is greater than or equal to 5 divided by the minimum of p and (1 p). Normal Approximation to Binomial (continued) n 5 min( p, 1 p) or (np, n (1 p)) 5 Normal Approximation to Binomial (continued) Binomial Distribution If n 5/min(p, q), then we can approximate the binomial distribution with a normal distribution with =np and 2=npq. Normal Approximation to Binomial Example 1 Suppose we toss a coin 25 times and we want to find the probability that we get at least 10 but not more than 15 heads. Normal Approximation to Binomial Example 1 (continued) Is n large enough? Normal Approximation to Binomial Example 1 (continued) Continuity Correction No area under the curve for X=10 or X=15 Assign interval under the curve from 9.5 to 15.5, so points 10, 11, 12, 13, 14, 15 are included. Normal Approximation to Binomial Example 1 (continued) Normal Approximation to Binomial Continuity Correction To make proper continuity correction, list # of successes included in the event The continuity correction allows us to calculate P(X=x) Binomial Probability P(21 < X 25) P(X = 10) P(5 X < 12) No. of Successes Normal Curve w/CC Normal Approximation to Binomial Example 2 Consider a lottery in which you have a ten percent chance of winning. Suppose you play 10,000 times. What is the probability of winning at least 1030 times? Normal Approximation to Binomial Example 2 (continued) Using Excel to Compute Standard Normal Probabilities Excel has two functions for computing probabilities and z values for a standard normal distribution NORMSDIST is used to compute the cumulative probability given a value of z NORMSINV is used to compute the z value given a cumulative probability Using Excel to Compute Standard Normal Probabilities Excel Formula Worksheet 1 2 3 4 5 6 7 8 9 A B Probabilities: Standard Normal Distribution P (z < 1.00) P (0.00 < z < 1.00) P (0.00 < z < 1.25) P (-1.00 < z < 1.00) P (z > 1.58) P (z < -0.50) =NORMSDIST(1) =NORMSDIST(1)-NORMSDIST(0) =NORMSDIST(1.25)-NORMSDIST(0) =NORMSDIST(1)-NORMSDIST(-1) =1-NORMSDIST(1.58) =NORMSDIST(-0.5) Using Excel to Compute Standard Normal Probabilities Excel Value Worksheet 1 2 3 4 5 6 7 8 9 A B Probabilities: Standard Normal Distribution P (z < 1.00) P (0.00 < z < 1.00) P (0.00 < z < 1.25) P (-1.00 < z < 1.00) P (z > 1.58) P (z < -0.50) 0.8413 0.3413 0.3944 0.6827 0.0571 0.3085 Using Excel to Compute Standard Normal Probabilities Excel Formula Worksheet 1 2 3 4 5 6 A B Finding z Values, Given Probabilities z value with .10 in upper tail z value with .025 in upper tail z value with .025 in lower tail =NORMSINV(0.9) =NORMSINV(0.975) =NORMSINV(0.025) Using Excel to Compute Standard Normal Probabilities Excel Value Worksheet 1 2 3 4 5 6 A B Finding z Values, Given Probabilities z value with .10 in upper tail z value with .025 in upper tail z value with .025 in lower tail 1.28 1.96 -1.96 Using Excel to Compute Normal Probabilities Excel has two functions for computing cumulative probabilities and x values for any normal distribution: NORMDIST is used to compute the cumulative probability given an x value NORMINV is used to compute the x value given a cumulative probability. Using Excel to Compute Normal Probabilities Excel Formula Worksheet 1 2 3 4 5 6 7 8 A B Probabilities: Normal Distribution P (x > 20) =1-NORMDIST(20,15,6,TRUE) Finding x Values, Given Probabilities x value with .05 in upper tail =NORMINV(0.95,15,6) Using Excel to Compute Normal Probabilities Excel Value Worksheet 1 2 3 4 5 6 7 8 A B Probabilities: Normal Distribution P (x > 20) 0.2023 Finding x Values, Given Probabilities x value with .05 in upper tail 24.87 Note: P(x > 20) = .2023 here using Excel, while our previous manual approach using the z table yielded .2033 due to our rounding of the z value.
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Biology 202 NAME: Fall 2007 T.A.: Homework 8 Discussion Section #: Due: Monday, October 29, by 5pm Turn in to T.A. mailbox 1. (3 pts) The human human pedigree on the next page shows people affected with the rare nail-patella syndrome (misshapen nail
New Mexico - BIO - 202
Biology 202 Fall 2007 Homework 9 Due: November 1st or 2ndNAME: Gaby morales T.A.: yadeeh escobedo Discussion Section #:91. (2 pts) You are given a strR (streptomycin resistant) Hfr strain and an ampR (ampicillin resistant) F strain. Even when the
New Mexico - BIO - 202
lac operon1. Hello! Sorry to hear about your kid. I was looking through the outlines and there were a few answers i couldnt find in the book or in our notes. I couldn't find info on the questions from study guide 11 about antibodies *I didn't talk
New Mexico - BIO - 202
Exam 3 Study Guide Linked genes genes on the same chromosome pair Linked genes violate independent assortment because they are not equally represented or chanced during meiosis. Cis arrangement parental type is PT and pt, recombinant is Pt and pT
New Mexico - BIO - 202
Bio 203L Homework 6: Ecology 1 (H6-10 points) Due Next Week (Mar 31st-Apr 4th, 2008)Spring 2008Section 1 (2 pts.): Locate two scientific journal articles that you expect to use for your Lemna/Salvinia project. You will then generate an annotated
New Mexico - BIO - 203
SI Work sheet # 6 Multiple Choice. 1. Choose the correct statements. a) Natural selection alone built the dog specific psychology of knowing when to avoid human beings. b) Artificial selection was the proximate of cause of a dog's ability to seek she
New Mexico - BIO - 203
1)Which statements are true regarding the good-of-the-species fallacy? a. It states that natural selection disfavors what is bad for the species. b. The good-of-the species is a great way to understand evolution. c. States what is seen in nature is
New Mexico - BIO - 203
1) Spiny lobsters spend the day hiding in cracks or holes in coral reefs. At night, they emerge and wander away from the reef in search of clams, muscles, or other sources of food. Before dawn, they use one of several dens they use on a regular basis
New Mexico - BIO - 203
SI worksheet # 4 Bio 203 Evolution Multiple Choice 1. Which of the following statements are true? a) Taxonomy is just another name for systematics. b) Taxonomy is the same as phylogeny c) Taxonomy reflects phylogeny d) Taxonomy is the classification
New Mexico - BIO - 203
SI Worksheet #5 Multiple Choice. 1. Which of the following statements are true? a) Darwin's Theory of life's history started becoming accepted in 1959. b) Darwin wrote his book The Origin of Species in 1859. c) Darwin states that life's history on ea
New Mexico - BIO - 203
Bio 203 SI worksheet #7 Multiple Choice. 1. Choose the correct statements. a) Fitness in biology is an individual's design for reproductive success. b) Big, symmetric peacock's tail is an example of high fitness. c) Small magnitude of gynoid fat in f
New Mexico - MATH - 181
Math 181, CALCULUS 2-Spring 2008Instructor: Office Hours: Calculator: TI83 Plus, or equivalent Office : Phone Number: Email:Text: Calculus and Its Applications, 11th Edition, by Goldstein, Lay, et al.Please note the following guidelines for the c
New Mexico - MUS - 139
Jounod in romantic. joined bach's chords from prelude and instead of playing one at a time played all notes together Countertenor higher than tenor. Tenor&lt;baritone&lt;bass Soprano is higher than alto Beginning of baroque we started having opera; Chinese
New Mexico - MUS - 139
MUS 139 Fall 2007 Exam 2 Review The exam will be multiple choice (please bring a pencil). The only dates you need to know are for the musical periods covered since exam 1: Baroque (1600-1750) Classical (1750-1825) Listening Examples: Be able to ident
New Mexico - MUS - 139
MUS 139.001 Fall 2007 Exam 3 Review Exam 3: December 12, 10:00 The exam will be multiple choice (please bring a pencil). The only dates you need to know are for the musical periods: Middle Ages (500-1450) Renaissance (1450-1600) Baroque (1600-1750) C
New Mexico - MUS - 139
10/23/2007 4:33:00 PM15 October, 2007 Hyden was the first true classical composer Beethoven is considered to be part of the classical period and beginning of the romantic Hyden o Most of his career was under a prince (Esterase (sp?) London Symphani
New Mexico - MUS - 139
24 September 2007 Baroque Freedom of expression Emotionality Exaggeration Extravagance o Lead to the development of theatrics Renaissance vs Baroque o Renaissance Voice ideal Voices used in ensembles A copella Natural, simple musical ideas Irreg
New Mexico - CHEM - 301
New Mexico - CHEM - 301
New Mexico - CHEM - 301
New Mexico - CHEM - 301
New Mexico - CHEM - 301
New Mexico - CHEM - 301
New Mexico - CHEM - 301
Name THIRD HOUR TESTKEY1. (21, 7 each) In the spaces provided below, give the correct IUPAC names for the compounds having each of the following structures.a. 7-chloro-3-cyano-5-cyclopentyl-2-ethoxy-6-oxohept-3-enoyl chlorideb. 5-amino-N-cycl
New Mexico - CHEM - 301
Name THIRD HOUR TESTKEY1. (21, 7 each) In the spaces provided below, give the correct IUPAC names for the compounds having each of the following structures.a. 7-chloro-3-cyano-5-cyclopentyl-2-ethoxy-6-oxohept-3-enoyl chlorideb. 5-amino-N-cycl