1. Chapter 5: Continuous Random Variables

2. McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
Where We’ve Been
Using probability rules to find the probability of discrete events
Examined probability models for discrete random variables
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

3. McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
Where We’re Going
Develop the notion of a probability distribution for a continuous random variable
Examine several important continuous random variables and their probability models
Introduce the normal probability distribution
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

4. 5.1: Continuous Probability Distributions
A continuous random variable can assume any numerical value within some interval or intervals.
The graph of the probability distribution is a smooth curve called a
probability density function,
frequency function or
probability distribution.
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

5. 5.1: Continuous Probability Distributions
There are an infinite number of possible outcomes
p(x) = 0
Instead, find p(a<x<b)
 Table
 Software
 Integral calculus)
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

6. 5.2: The Uniform Distribution
X can take on any value between c and d with equal probability
= 1/(d - c)
For two values a and b
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

7. 5.2: The Uniform Distribution
Mean:
Standard Deviation:
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

8. 5.2: The Uniform Distribution
Suppose a random variable x is distributed uniformly with c = 5 and d = 25. What is P(10  x  18)?
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

9. 5.2: The Uniform Distribution
Suppose a random variable x is distributed uniformly with c = 5 and d = 25. What is P(10  x  18)?
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

10. 5.3: The Normal Distribution
Closely approximates many situations
Perfectly symmetrical around its mean
The probability density function f(x):
µ = the mean of x
 = the standard deviation of x
 = …
e = …
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

11. 5.3: The Normal Distribution
Each combination of µ and  produces a unique normal curve
The standard normal curve is used in practice, based on the standard normal random variable z (µ = 0,  = 1), with the probability distribution
The probabilities for z are given in Table IV
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

12. 5.3: The Normal Distribution
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

13. 5.3: The Normal Distribution
So any normally distributed variable can be analyzed with this single distribution
For a normally distributed random variable x, if we know µ and ,
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

14. 5.3: The Normal Distribution
Say a toy car goes an average of 3,000 yards between recharges, with a standard deviation of 50 yards (i.e., µ = 3,000 and  = 50)
What is the probability that the car will go more than 3,100 yards without recharging?
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

15. 5.3: The Normal Distribution
Say a toy car goes an average of 3,000 yards between recharges, with a standard deviation of 50 yards (i.e., µ = 3,000 and  = 50)
What is the probability that the car will go more than 3,100 yards without recharging?
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

16. 5.3: The Normal Distribution
To find the probability for a normal random variable …
Sketch the normal distribution
Indicate x’s mean
Convert the x variables into z values
Put both sets of values on the sketch, z below x
Use Table IV to find the desired probabilities
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

17. 5.4: Descriptive Methods for Assessing Normality
If the data are normal
A histogram or stem-and-leaf display will look like the normal curve
The mean ± s, 2s and 3s will approximate the empirical rule percentages
The ratio of the interquartile range to the standard deviation will be about 1.3
A normal probability plot , a scatterplot with the ranked data on one axis and the expected z-scores from a standard normal distribution on the other axis, will produce close to a straight line
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

18. 5.4: Descriptive Methods for Assessing Normality
Errors per MLB team in 2003
Mean: 106
Standard Deviation: 17
IQR: 22  
22 out of 30: 73%
28 out of 30: 93%
30 out of 30: 100%
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

19. 5.4: Descriptive Methods for Assessing Normality
A normal probability plot is a scatterplot with the ranked data on one axis and the expected z-scores from a standard normal distribution on the other axis
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

20. McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.5: Approximating a Binomial Distribution with the Normal Distribution
Discrete calculations may become very cumbersome
The normal distribution may be used to approximate discrete distributions
The larger n is, and the closer p is to .5, the better the approximation
Since we need a range, not a value, the correction for continuity must be used
A number r becomes r+.5
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

21. McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.5: Approximating a Binomial Distribution with the Normal Distribution
Calculate the mean plus/minus 3 standard deviations
If this interval is in the range 0 to n,
the approximation will be reasonably close
Express the binomial probability as a range of values
Find the z-values for each binomial value
Use the standard normal distribution to find
the probability for the range of values you calculated
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

22. McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.5: Approximating a Binomial Distribution with the Normal Distribution
Flip a coin 100 times and compare the binomial and normal results
Binomial:Normal:
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

23. McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.5: Approximating a Binomial Distribution with the Normal Distribution
Flip a weighted coin [P(H)=.4] 10 times and compare the results
Binomial:Normal:
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

24. McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables
5.5: Approximating a Binomial Distribution with the Normal Distribution
Flip a weighted coin [P(H)=.4] 10 times and compare the results
Binomial:Normal:
The more p differs from .5,
and the smaller n is,
the less precise the
approximation will be
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

25. 5.6: The Exponential Distribution
Probability Distribution for an Exponential Random Variable x
Probability Density Function
Mean: µ = 
Standard Deviation:  = 
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

26. 5.6: The Exponential Distribution
Suppose the waiting time to see the nurse at the student health center is distributed exponentially with a mean of 45 minutes. What is the probability that a student will wait more than an hour to get his or her generic pill? 60
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables

27. 5.6: The Exponential Distribution
Suppose the waiting time to see the nurse at the student health center is distributed exponentially with a mean of 45 minutes. What is the probability that a student will wait more than an hour to get his or her generic pill? 60
McClave: Statistics, 11th ed. Chapter 5: Continuous Random Variables