Chapter 4 Continuous Random Variables and Probability Distributions
Chapter 4 Continuous Random Variables and Probability Distributions § 4. 1 - Probability Density Functions § 4. 2 - Cumulative Distribution Functions and Expected Values § 4. 3 - The Normal Distribution § 4. 4 - The Exponential and Gamma Distributions § 4. 5 - Other Continuous Distributions § 4. 6 - Probability Plots
~ The Normal Distribution ~ (a. k. a. “The Bell Curve”) standard deviation X ~ N(μ, σ) σ Johann Carl Friedrich Gauss 1777 -1855 X mean μ • Symmetric, unimodal • Models many (but not all) natural systems • Mathematical properties make it useful to work with 2
L A I SPEC E CAS Standard Normal Distribution Z ~ N(0, 1) 1 Total Area = 1 Z The cumulative distribution function (cdf) is denoted by (z). It is not expressible in explicit, closed form, but is tabulated, and computable in R via the command pnorm.
Example Standard Normal Distribution Find (1. 2) = P(Z 1. 2). Z ~ N(0, 1) 1 Total Area = 1 Z 1. 2 “z-score”
Example Standard Normal Distribution Find (1. 2) = P(Z 1. 2). Z ~ N(0, 1) Ø Use the included table. 1 Total Area = 1 Z 1. 2 “z-score”
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Example Standard Normal Distribution Find (1. 2) = P(Z 1. 2). Z ~ N(0, 1) Ø Use the included table. Ø Use R: > pnorm(1. 2) [1] 0. 8849303 1 Total Area = 1 0. 88493 P(Z > 1. 2) 0. 11507 Z 1. 2 “z-score” 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.
Standard Normal Distribution X ~ N(μ, σ) Z ~ N(0, 1) σ μ 1 Z Why be concerned about this, when most “bell curves” don’t have mean = 0, and standard deviation = 1? Any normal distribution can be transformed to the standard normal distribution via a simple change of variable.
Example POPULATION Random Variable X = Age at first birth 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) σ = 1. 5 μ = 25. 4 27. 2 10
Example POPULATION Random Variable X = Age at first birth 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) The x-score = 27. 2 must first be transformed to a corresponding z-score. σ = 1. 5 μ μ==25. 4 27. 2 33 11
Example POPULATION Random Variable X = Age at first birth Question: What proportion of the population had their first child before the age of 27. 2 years old? P(X < 27. 2) = ? P(Z < 1. 2) = 0. 88493 Year 2010 X ~ N(25. 4, 1. 5) σ = 1. 5 Ø Using R: > pnorm(27. 2, 25. 4, 1. 5) [1] 0. 8849303 μ μ==25. 4 27. 2 33 12
Standard Normal Distribution Z ~ N(0, 1) 1 Z What symmetric interval about the mean 0 contains 95% of the population values? That is…
Standard Normal Distribution Z ~ N(0, 1) Ø Use the included table. 0. 95 0. 025 Z -z. 025 = ? +z. 025 = ? What symmetric interval about the mean 0 contains 95% of the population values? That is…
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Standard Normal Distribution Z ~ N(0, 1) Ø Use the included table. Ø Use R: > qnorm(. 025) [1] -1. 959964 > qnorm(. 975) [1] 1. 959964 0. 95 0. 025 Z -z. 025 ? . 025 = -1. 96 “. 025 critical values” +z. 025 ? . 025 = +1. 96 What symmetric interval about the mean 0 contains 95% of the population values?
X ~ N(μ 1. 5) , σ) X ~ N(25. 4, Standard Normal Distribution Z ~ N(0, 1) What symmetric interval about the mean age of 25. 4 contains 95% of the population values? 22. 46 X 28. 34 yrs > areas = c(. 025, . 975) > qnorm(areas, 25. 4, 1. 5) [1] 22. 46005 28. 33995 0. 025 Z -z. 025 ? . 025 = -1. 96 “. 025 critical values” +z. 025 ? . 025 = +1. 96 What symmetric interval about the mean 0 contains 95% of the population values?
Standard Normal Distribution Z ~ N(0, 1) Ø Use the included table. 0. 90 0. 05 Z Similarly… -z. 05 = ? +z. 05 = ? What symmetric interval about the mean 0 contains 90% of the population values?
…so average 1. 64 and 1. 65 0. 95 average of 0. 94950 and 0. 95053… 20
Standard Normal Distribution Z ~ N(0, 1) Ø Use the included table. Ø Use R: > qnorm(. 05) [1] -1. 644854 > qnorm(. 95) [1] 1. 644854 0. 90 0. 05 Z Similarly… -z. 05 = -1. 645 ? “. 05 critical values” +z +z. 05 = +1. 645 ? . 05 = What symmetric interval about the mean 0 contains 90% of the population values?
Standard Normal Distribution Z ~ N(0, 1) In general…. 10. 90 – 0. 05 /2 Z Similarly… -z. 05 = -1. 645 ? -z / 2 ““. 05 / 2 criticalvalues” +z +z. 05 = +1. 645 ? . 05 / 2= What symmetric interval about the mean 0 contains 100(1 – )% of the population values?
continuous discrete Normal Approximation to the Binomial Distribution Suppose a certain outcome exists in a population, with constant probability . 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) P(Success) = P(Failure) = 1 – Then X is said to follow a Binomial distribution, written X ~ Bin(n, ), with “probability function” p(x) = , x = 0, 1, 2, …, n. 23
> dbinom(10, 100, . 2) [1] 0. 00336282 Area 24
> pbinom(10, 100, . 2) [1] 0. 005696381 Area 25
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Therefore, if… X ~ Bin(n, ) with n 15 and n (1 – ) 15, then… That is… “Sampling Distribution” of 30
Classical Continuous Probability Distributions ● Normal distribution ● Exponential distribution ● Log-Normal ~ X is not normally distributed (e. g. , skewed), but Y = “logarithm of X” is normally distributed ● Student’s 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…
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