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# Statr sessions 9 to 10

## on Dec 04, 2013

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## Statr sessions 9 to 10Presentation Transcript

• Discrete Distributions • What is a random variable? • Distinguish between discrete random variables and continuous random variables. • Know how to determine the mean and variance of a discrete distribution. • Identify the type of statistical experiments that can be described by the binomial distribution, and know how to calculate probabilities based on the binomial distribution.
• Discrete vs. Continuous Distributions • Random Variable - a variable which contains the outcomes of a chance experiment • Discrete Random Variable – A random variable that only takes on distinct values ex: Number of heads on 10 flips, Number of defective items in a random sample of 100, Number of times you check your watch during class, etc. • Continuous Random Variable – A random variable that takes on infinite values by increasing precision. For each two values, there always exists a valid value in between them. ex: Time until a bulb goes out, height, etc.
• Describing a Distribution • A distribution can be described by constructing a graph of the distribution • Measures of central tendency and variability can be applied to distributions
• Describing a Discrete Distribution • Mean of discrete distribution – is the long run average – If the process is repeated long enough, the average of the outcomes will approach the long run average (mean) – Mean of a discrete distribution µ = ∑ (Xi * P(Xi)) where µ is the long run average, Xi = the ith outcome of random variable X, and P(Xi) = probability of X = Xi
• Describing a Discrete Distribution • Variance of a discrete distribution is obtained in a manner similar to raw data, summing the squared deviations from the mean and weighting them by P(Xi) (rather than dividing by n): Var(Xi) = ∑ (Xi – m)2* P(Xi) • Standard Deviation is computed by taking the square root of the variance
• Discrete Distribution -- Example • An executive is considering out-of-town business travel for a given Friday. At least one crisis could occur on the day that the executive is gone. The distribution on the following slide contains the number of crises that could occur during the day the executive is gone and the probability that each number will occur. For example, there is a 0.37 probability that no crisis will occur, a 0.31 probability of one crisis, and so on.
• Discrete Distribution -- Example Distribution of Daily Crises Number of Crises ( X ) 0 1 2 3 4 5 Probability P(Xi) 0.37 0.31 0.18 0.09 0.04 0.01 P r o b a b i l i t y P(Xi) 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 Number of Crises ( X ) 5
• Requirements for a Discrete Probability Function -- Examples • Each probability must be between 0 and 1 • The sum of all probabilities must be equal to 1. X P(X) X P(X) X P(X) -1 0 1 2 3 .1 .2 .4 .2 .1 1.0 -1 0 1 2 3 -.1 .3 .4 .3 .1 1.0 -1 0 1 2 3 .1 .3 .4 .3 .1 1.2 VALID NOT VALID NOT VALID
• Roulette A roulette wheel has 37 pockets. £1 on a number returns £36 if it comes up (i.e. your £1 back + £35 winnings). Otherwise you lose your £1. What is the expected winnings (in pounds) on a £1 number bet? 1. 2. 3. 4. 5. -1/36 -1/37 -2/37 -1/35 1/36
• Roulette A roulette wheel has 37 pockets. £1 on a number returns £36 if it comes up (i.e. your £1 back + £35 winnings). Otherwise you lose your £1. What is the expected winnings (in pounds) on a £1 number bet?
• Binomial Distribution • The binomial distribution is a discrete distribution where X, the random variable, represents the number of “successes” and the following four conditions are met:     There are n trials The n trials are independent of each other The outcome is dichotomous – only two outcomes are possible The probability of “success” is constant across trials • Example, 10 coin flips, X = # of heads • X = the number of “successes” and we say X follows a Binomial distribution with n trials and P(success in each trial) = p • If the data follow a binomial distribution, then we can summarize P(Xi) for all values of Xi = 1, …, n through the binomial probability distribution formula • n = Sample size
• Situations where a Binomial distribution might occur 1) Quality control: select n items at random; X = number found to be satisfactory. 2) Survey of n people about products A and B; X = number preferring A. 3) Telecommunications: n messages; X = number with an invalid address. 4) Number of items with some property above a threshold; e.g. X = number with height > A
• Binomial distribution • Probability function • Mean value • Variance and Standard Deviation n! P( X )  p X  qn  X X !n  X ! for 0  X  n, q  1  p m  n p  2  n pq    2  n pq
• Binomial Distribution: Demonstration Problem 5.3 According to the U.S. Census Bureau, approximately 6% of all workers in Jackson, Mississippi, are unemployed. In conducting a random telephone survey in Jackson, what is the probability of getting two or fewer unemployed workers in a sample of 20?
• Binomial Distribution: Demonstration Problem 5.3 According to the U.S. Census Bureau, approximately 6% of all workers in Jackson, Mississippi, are unemployed. In conducting a random telephone survey in Jackson, what is the probability of getting two or fewer unemployed workers in a sample of 20? • In this example, – 6% are unemployed => p – The sample size is 20 => n – 94% are employed => q – X is the number of successes desired – What is the probability of getting 2 or fewer unemployed workers in the sample of 20? => P(X≤2) – The hard part of this problem is identifying p, n, and x
• Binomial Distribution Table: Demonstration Problem 5.3 According to the U.S. Census Bureau, approximately 6% of all workers in Jackson, Mississippi, are unemployed. In conducting a random telephone survey in Jackson, what is the probability of getting two or fewer unemployed workers in a sample of 20? n = 20 PROBABILITY X 0.05 0.06 0.07 0 0.3585 0.2901 0.2342 1 0.3774 0.3703 0.3526 2 0.1887 0.2246 0.2521 3 0.0596 0.0860 0.1139 4 0.0133 0.0233 0.0364 5 0.0022 0.0048 0.0088 6 0.0003 0.0008 0.0017 7 0.0000 0.0001 0.0002 8 0.0000 0.0000 0.0000 … … … … 20 0.0000 0.0000 0.0000 n  20 p  .06 q  .94 P( X  2)  P( X  0)  P( X  1)  P( X  2)  .2901 .3703  .2246  .8850
• Poisson Distribution • The Poisson distribution focuses only on the number of discrete occurrences over some interval or continuum  Poisson does not have a given number of trials (n) as a binomial experiment does  Occurrences are independent of other occurrences  Occurrences occur over an interval
• Poisson Distribution • If Poisson distribution is studied over a long period of time, a long run average can be determined  The average is denoted by lambda (λ)  Each Poisson distribution contains a lambda value from which the probabilities are determined  A Poisson distribution can be described by λ alone
• Poisson Distribution : Probability Function P(x)  X e   for X  0,1,2,3,... P( X )  X! where :   longrun average e  2.718282... (the base of natural logarithms) Mean Variance   Standard Deviation 
• Continuous Random Variables • A continuous random variable is a random variable which can take values measured on a continuous scale e.g. weights, strengths, times or lengths. • Probabilities of outcomes occurring between particular two points are determined by calculating the area under the Probability density function curve between these points.
• Properties of Normal Distribution • • • • • Continuous distribution - Line does not break The line does not touch the x-axis Bell-shaped, symmetrical distribution Ranges from -∞ to ∞ Mean = median = mode • • Area under the curve = total probability = 1 68% of data are within one standard deviation of mean, 95% within two standard deviations, and 99.7% within three standard deviations by Empirical rule.
• Probability Density Function of Normal Distribution There are a number of different normal distributions, they are characterized by the mean and the standard deviation
• Probability Density Function of Normal Distribution 1 x m  1    2 f ( x)  e     2 where : m  mean of x   standard deviation of x  = 3.14159. . . e  2.71828. . . 2 m
• Normal Distribution – Calculating Probabilities • Rather than create a different table for every normal distribution (with different mean and standard deviations), we can calculate a standardized normal distribution, called Z-score • A z-score gives the number of standard deviations that a value x is above the mean. • Z distribution is normal distribution with a mean of 0 and a standard deviation of 1
• Standardized Normal Distribution – Calculating Probabilities • Z distribution probability values are given in table A5 of your book or can be calculated using software • Table A5 gives the total area under the Z curve between 0 and any point on the positive Z axis • Since the curve is symmetric, the area under the curve between Z and 0 is the same whether the Z curve is positive or negative
• Standardized Normal Distribution – Calculating Probabilities – z table Second Decimal Place in z z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.00 0.10 0.20 0.30 0.0000 0.0398 0.0793 0.1179 0.0040 0.0438 0.0832 0.1217 0.0080 0.0478 0.0871 0.1255 0.0120 0.0517 0.0910 0.1293 0.0160 0.0557 0.0948 0.1331 0.0199 0.0596 0.0987 0.1368 0.0239 0.0636 0.1026 0.1406 0.0279 0.0675 0.1064 0.1443 0.0319 0.0714 0.1103 0.1480 0.0359 0.0753 0.1141 0.1517 0.90 1.00 1.10 1.20 0.3159 0.3413 0.3643 0.3849 0.3186 0.3438 0.3665 0.3869 0.3212 0.3461 0.3686 0.3888 0.3238 0.3485 0.3708 0.3907 0.3264 0.3508 0.3729 0.3925 0.3289 0.3531 0.3749 0.3944 0.3315 0.3554 0.3770 0.3962 0.3340 0.3577 0.3790 0.3980 0.3365 0.3599 0.3810 0.3997 0.3389 0.3621 0.3830 0.4015 2.00 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 3.00 3.40 3.50 0.4987 0.4997 0.4998 0.4987 0.4997 0.4998 0.4987 0.4997 0.4998 0.4988 0.4997 0.4998 0.4988 0.4997 0.4998 0.4989 0.4997 0.4998 0.4989 0.4997 0.4998 0.4989 0.4997 0.4998 0.4990 0.4997 0.4998 0.4990 0.4998 0.4998
• Table Lookup of a Standardized Normal Probability P(0  Z  1)  0. 3413 Z 0.00 0.01 0.02 0.00 0.10 0.20 1.00 -3 -2 -1 0 1 2 3 0.0000 0.0040 0.0080 0.0398 0.0438 0.0478 0.0793 0.0832 0.0871 0.3413 0.3438 0.3461 1.10 1.20 0.3643 0.3665 0.3686 0.3849 0.3869 0.3888
• Applying the Z Formula X is normally distributed withm = 485, and  = 105 P( 485  X  600)  P( 0  Z  1.10) . 3643 For X = 485, X - m 485  485 Z=  0  105 For X = 600, X - m 600  485 Z=   1.10  105 Z 0.00 0.01 0.02 0.00 0.10 0.0000 0.0040 0.0080 0.0398 0.0438 0.0478 1.00 0.3413 0.3438 0.3461 1.10 0.3643 0.3665 0.3686 1.20 0.3849 0.3869 0.3888
• Applying the Z Formula X is normally distributed with m = 494, and  = 100 P( X  550 )  P( Z  0.56 )  .7123 For X = 550 X - m 550  494 Z=   0.56  100 0.5 + 0.2123 = 0.7123
• Applying the Z Formula X is normally distributed with m = 494, and  = 100 P( X  700 )  P ( Z  2.06 )  .0197 For X = 700 Z= X - m 700  494   2.06  100 0.5 – 0.4803 = 0.0197
• Applying the Z Formula X is normally distributed with m = 494, and  = 100 P(300  X  600 )  P (1.94  Z  1.06 )  .8292 For X = 300 Z= X-m  300  494   1.94 100 For X = 600 Z= X-m  600  494   1.06 100 0.4738+ 0.3554 = 0.8292