1. 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

2. X is a linear function of Z
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 ~ N(0, 1)
X is a linear function of Z

3. X is a linear function of Z
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
IF our data approximates a bell curve, then its quantiles should “line up” with those of N(0, 1).
Z ~ N(0, 1)
X is a linear function of Z
Q-Q plot
Normal scores plot
Normal probability plot

4. X is a linear function of Z
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…)

5. And what do we do if it’s not, or we can’t tell?
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

6. And what do we do if it’s not, or we can’t tell?
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.”

7. And what do we do if it’s not, or we can’t tell?
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,…).
qqnorm(x, pch = 19, cex = .5)
qqline(x)
qqnorm(y, pch = 19, cex = .5)
qqline(y)

8. 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

9. 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,…).