Download 4.6

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
X ~ N ( ,  )
How do we check that this assumption is
reasonable, when all we have is a sample?
{ x1 , x2 , x3 ,…, x24 }
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).
Z
X 

 X    Z
X is a linear function of Z
X ~ N ( ,  )
How do we check that this assumption is
reasonable, when all we have is a sample?
{ x1 , x2 , x3 ,…, x24 }
And what do we do if it’s not, or we can’t tell?
Sample quantiles
Z ~ N(0, 1)
IF our data approximates a bell
curve, then its quantiles should
“line up” with those of N(0, 1).
Z
X 

 X    Z
X is a linear function of Z
• Q-Q plot
• Normal scores plot
• Normal probability plot
X ~ N ( ,  )
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).
Z
X 

 X    Z
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…)
X ~ N ( ,  )
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).
Z
X 

 X    Z
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
X ~ N ( ,  )
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?
 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?
X ~ N ( ,  )
And what do we do if it’s not, or we can’t tell?
 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)
X ~ N ( ,  )
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?
 Use a mathematical “transformation” of the data (e.g., log, square root,…).
Cauchy distribution
X ~ N ( ,  )
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?
 Use a mathematical “transformation” of the data (e.g., log, square root,…).