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
How do we check that this assumption is reasonable, when all we have is a sample? And what do we do if it’s not, or we can’t tell? Z ~ N(0, 1) IF our data approximates a bell curve, then its quantiles should “line up” with those of N(0, 1). X is a linear function of Z qnorm(. 04*1: 24)
How do we check that this assumption is reasonable, when all we have is a sample? And what do we do if it’s not, or we can’t tell? Sample quantiles Z ~ N(0, 1) qnorm(. 04*1: 24) IF our data approximates a bell curve, then its quantiles should “line up” with those of N(0, 1). X is a linear function of Z • Q-Q plot • Normal scores plot • Normal probability plot
How do we check that this assumption is reasonable, when all we have is a sample? And what do we do if it’s not, or we can’t tell? IF our data approximates a bell curve, then its quantiles should “line up” with those of N(0, 1). X is a linear function of Z • Q-Q plot • Normal scores plot • Normal probability plot qqnorm(mysample) qqline(mysample) (R uses a slight variation to generate quantiles…)
How do we check that this assumption is reasonable, when all we have is a sample? And what do we do if it’s not, or we can’t tell? IF our data approximates a bell curve, then its quantiles should “line up” with those of N(0, 1). X is a linear function of Z • Q-Q plot • Normal scores plot • Normal probability plot qqnorm(mysample) qqline(mysample) (R uses a slight variation to generate quantiles…) Formal statistical tests exist; see notes. Method can be extended to other models
How do we check that this assumption is reasonable, when all we have is a sample? And what do we do if it’s not, or we can’t tell? Ø Use a mathematical “transformation” of the data (e. g. , log, square root, …). x = rchisq(1000, 15) hist(x) y = log(x) hist(y) X is said to be “log-normal. ”
How do we check that this assumption is reasonable, when all we have is a sample? And what do we do if it’s not, or we can’t tell? Ø Use a mathematical “transformation” of the data (e. g. , log, square root, …). qqnorm(x, pch = 19, cex =. 5) qqline(x) qqnorm(y, pch = 19, cex =. 5) qqline(y)
How do we check that this assumption is reasonable, when all we have is a sample? And what do we do if it’s not, or we can’t tell? Ø Use a mathematical “transformation” of the data (e. g. , log, square root, …). Cauchy distribution
How do we check that this assumption is reasonable, when all we have is a sample? And what do we do if it’s not, or we can’t tell? Ø Use a mathematical “transformation” of the data (e. g. , log, square root, …).
- Slides: 9