CHAPTER 4 4 4.1 - Discrete Models ïƒ G eneral distributions ïƒ C lassical:...

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Transcript of CHAPTER 4 4 4.1 - Discrete Models ïƒ G eneral distributions ïƒ C lassical:...

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CHAPTER 4 4 4.1 - Discrete Models G eneral distributions C lassical: Binomial, Poisson, etc. 4 4.2 - Continuous Models G eneral distributions C lassical: Normal, etc. Slide 2 X 2 ~ The Normal Distribution ~ (a.k.a. The Bell Curve) Johann Carl Friedrich Gauss 1777-1855 mean standard deviation X ~ N( , ) Symmetric, unimodal Models many (but not all) natural systems Mathematical properties make it useful to work with Slide 3 Standard Normal Distribution Z ~ Z ~ N(0, 1) Z The cumulative distribution function (cdf) is denoted by ( z ). It is tabulated, and computable in R via the command pnorm. SPECIAL CASE Total Area = 1 1 Slide 4 Z 1 Standard Normal Distribution Z ~ Z ~ N(0, 1) Example Find P(Z 1.2). 1.2 z-score Total Area = 1 Slide 5 Z Standard Normal Distribution Z ~ Z ~ N(0, 1) Example Find P(Z 1.2). 1 1.2 Use the included table. z-score Total Area = 1 Slide 6 6 Lecture Notes Appendix Slide 7 7 Slide 8 Z Standard Normal Distribution Z ~ Z ~ N(0, 1) Example Find P(Z 1.2). 1 1.2 Use the included table. 0.88493 Use R: > pnorm(1.2) [1] 0.8849303 z-score P(Z > 1.2) 0.11507 Total Area = 1 Note: Because this is a continuous distribution, P(Z = 1.2) = 0, so there is no difference between P(Z > 1.2) and P(Z 1.2), etc. Slide 9 Standard Normal Distribution Z ~ Z ~ N(0, 1) Z X ~ N( , ) 1 Any normal distribution can be transformed to the standard normal distribution via a simple change of variable. Why be concerned about this, when most bell curves dont have mean = 0, and standard deviation = 1? Slide 10 Year 2010 X ~ N(25.4, 1.5) = 25.4 = 1.5 10 Example Random Variable X = Age at first birth POPULATION Question: What proportion of the population had their first child before the age of 27.2 years old? P(X < 27.2) = ? 27.2 Slide 11 11 Example Random Variable X = Age at first birth POPULATION Question: What proportion of the population had their first child before the age of 27.2 years old? P(X < 27.2) = ? Year 2010 X ~ N(25.4, 1.5) = 1.5 = = 33 The x-score = 27.2 must first be transformed to a corresponding z-score. = 25.4 27.2 Slide 12 12 Example Random Variable X = Age at first birth POPULATION Question: What proportion of the population had their first child before the age of 27.2 years old? P(X < 27.2) = ? = 1.5 = = 33 P(Z < 1.2) = 0.88493 Using R: > pnorm(27.2, 25.4, 1.5) [1] 0.8849303 Year 2010 X ~ N(25.4, 1.5) = 25.427.2 Slide 13 Z What symmetric interval about the mean 0 contains 95% of the population values? That is 1 Standard Normal Distribution Z ~ Z ~ N(0, 1) Slide 14 Z 0.95 0.025 +z.025 = ? -z.025 = ? What symmetric interval about the mean 0 contains 95% of the population values? That is Standard Normal Distribution Z ~ Z ~ N(0, 1) Use the included table. Slide 15 15 Lecture Notes Appendix Slide 16 16 Slide 17 Use the included table. +z.025 = ?+z.025 = +1.96-z.025 = ? Standard Normal Distribution Z ~ Z ~ N(0, 1) Z 0.95 0.025 What symmetric interval about the mean 0 contains 95% of the population values? -z.025 = -1.96 .025 critical values Use R: > qnorm(.025) [1] -1.959964 > qnorm(.975) [1] 1.959964 Slide 18 +z.025 = ?+z.025 = +1.96-z.025 = ? Standard Normal Distribution Z ~ Z ~ N(0, 1) Z 0.95 0.025 What symmetric interval about the mean 0 contains 95% of the population values? -z.025 = -1.96 .025 critical values What symmetric interval about the mean age of 25.4 contains 95% of the population values? 22.46 X 28.34 yrs X ~ N( , ) X ~ N(25.4, 1.5) > areas = c(.025,.975) > qnorm(areas, 25.4, 1.5) [1] 22.46005 28.33995 Slide 19 Use the included table. Standard Normal Distribution Z ~ Z ~ N(0, 1) Z 0.90 0.05 +z.05 = ?-z.05 = ? What symmetric interval about the mean 0 contains 90% of the population values? Similarly Slide 20 20 so average 1.64 and 1.65 0.95 average of 0.94950 and 0.95053 Slide 21 Use the included table. -z.05 = ?-z.05 = -1.645 Standard Normal Distribution Z ~ Z ~ N(0, 1) Z 0.90 0.05 +z.05 = ? What symmetric interval about the mean 0 contains 90% of the population values? Similarly +z.05 = +1.645 .05 critical values Use R: > qnorm(.05) [1] -1.644854 > qnorm(.95) [1] 1.644854 Slide 22 -z.05 = ?-z.05 = -1.645 Standard Normal Distribution Z ~ Z ~ N(0, 1) Z 0.90 0.05 +z.05 = ? What symmetric interval about the mean 0 contains 100(1 )% of the population values? Similarly +z.05 = +1.645 .05 critical values In general. 1 / 2 -z / 2 +z / 2 / 2 critical values Slide 23 23 continuousdiscrete Suppose a certain outcome exists in a population, with constant probability . P(Success) = P(Failure) = 1 We will randomly select a random sample of n individuals, so that the binary Success vs. Failure outcome of any individual is independent of the binary outcome of any other individual, i.e., n Bernoulli trials (e.g., coin tosses). Discrete random variable X = # Successes in sample (0, 1, 2, 3, ,, n) Discrete random variable X = # Successes in sample (0, 1, 2, 3, ,, n) Then X is said to follow a Binomial distribution, written X ~ Bin(n, ), with probability function f(x) =, x = 0, 1, 2, , n. Slide 24 24 > dbinom(10, 100,.2) [1] 0.00336282 Area Slide 25 25 > pbinom(10, 100,.2) [1] 0.005696381 Area Slide 26 26 Slide 27 27 Slide 28 28 Slide 29 29 Slide 30 30 Therefore, if X ~ Bin( n, ) with n 15 and n (1 ) 15, then Therefore, if X ~ Bin( n, ) with n 15 and n (1 ) 15, then That is Sampling Distribution of Slide 31 31 Normal distribution Log-Normal ~ X is not normally distributed (e.g., skewed), but Y = logarithm of X is normally distributed Students t-distribution ~ Similar to normal distr, more flexible F-distribution ~ Used when comparing multiple group means Chi-squared distribution ~ Used extensively in categorical data analysis Others for specialized applications ~ Gamma, Beta, Weibull