1. Unit 5: Modelling Continuous Data
Lesson 1: Continuous Probability Distributions
Introduction
The binomial distribution from last unit dealt with discrete random variables, which were always represented with a whole number (e.g. number of heads.)
In this unit we will be looking at continuous variables where the value can be displayed as fractions or decimals (e.g. the height in centimetres)
Lesson 1:
Continuous
Probability
Distribution

2. Unit 8: The Normal Distribution
Lesson 1: Continuous Probability Distributions
Discrete Variables –
Binomial Distribution
The number of head tossed
when a coin is tossed 5 times
Continuous Variables –
Normal Distribution
Assembly times for 36 tricycles
How can we make the histogram
Continuous?
Lesson 1:
Continuous
Probability
Distribution

3. Unit 8: The Normal Distribution
Lesson 1: Continuous Probability Distributions
Continuous Probability Distributions
As stated above the variables that we will focus on are those of the continuous nature
Let us now examine a special case:
Investigate and Inquire
The table below gives the failure rates for a model of computer printer during its first four years of use.
Lesson 1:
Continuous
Probability
Distribution
Age of Printer (Months)
Failure Rate (%)
0-6
4.5
6-12
2.4
12-18
1.2
18-24
1.5
24-30
2.2
30-36
3.1
36-42
4.0
42-48
5.8

4. Unit 8: The Normal Distribution
Lesson 1: Continuous Probability Distributions
Lesson 1:
Continuous
Probability
Distribution
Construct a scatter plot of these data. Use the midpoint of each interval. Sketch a smooth curve that is good fit to the data
Why do you think printer failure rates would have this shape of distribution?
Calculate the mean and the standard deviation of the failure rates. How useful are these summary measures in this distribution?

5. Unit 8: The Normal Distribution
Lesson 1: Continuous Probability Distributions
Calculating Standard Deviation and Mean:
Age of Printer (x) (Months)
Failure Rate (%)
X -Mean
(x-x)2
3.5
4.5
-21
441
9.5
2.4
-15
225
15.5
1.2 -9 81
21.5
1.5 -3 9
27.5
2.2 3
33.5
3.1
39.5 4 15
45.5
5.8 21
Mean=24.5
Σ =1512
Lesson 1:
Continuous
Probability
Distribution

6. Unit 8: The Normal Distribution
Lesson 1: Continuous Probability Distributions
The mean printer life is 24 months, with a standard deviation of 14.7 months. These summary measures are not very useful in describing the distribution, since so many data points lie far from the mean.
How can this data be represented?
Using a continuous curve of best fit.
Lesson 1:
Continuous
Probability
Distribution

7. Unit 8: The Normal Distribution
Lesson 1: Continuous Probability Distributions
Terminology of Continuous Distributions
Positively Skewed – A distribution with no symmetry and is pulled to the right
Lesson 1:
Continuous
Probability
Distribution
Negative Skewed – A distribution with no symmetry and is pulled to the left

8. Unit 8: The Normal Distribution
Lesson 1: Continuous Probability Distributions
Unimodal – The examples above are unimodal since they only contain one “hump”. This is similar to the mode of a set of discrete values.
Bimodal - Contains two “humps”. For example the speed in the 100 m for Olympic athletes would have two “humps” since it would contain different times for males and females.
Lesson 1:
Continuous
Probability
Distribution

9. Unit 8: The Normal Distribution
Lesson 1: Continuous Probability Distributions
Example 1-Uniform Continuous Distributions
Suppose the commuting time form Georgetown to downtown Toronto varies uniformly from 30 to 55 minutes, depending on traffic and weather conditions. Construct a graph of this distribution and use the graph to find:
The probability that a trip takes 45 minutes or less.
The probability that a trip takes more than 48 minutes.
Lesson 1:
Continuous
Probability
Distribution
We know that every time from 30 to 55 min is equally likely to occur
The distribution would be a horizontal line since this is uniform
The area under the line must equal 1 since each time is equally likely to occur and the total probability is 1.
The probability of each event to occur can be found by its interval:
This means that the likelihood that you would arrive in Toronto at any time from 30 – 55 minutes is 4%.

10. Unit 8: The Normal Distribution
Lesson 1: Continuous Probability Distributions 30 55
0.04
Minutes
Prob. Density
The distribution graph is a uniform one
Lesson 1:
Continuous
Probability
Distribution
a) The graph for less than 45 minutes would look like the yellow shade below: 30 55
0.04
Minutes
Prob. Density 45
P(Time<45) = 0.04(45-30)
=0.6 or 60%

11. Unit 8: The Normal Distribution
Lesson 1: Continuous Probability Distributions
b) The graph for greater than 48 minutes would look like the yellow shade below: 30 55
0.04
Minutes
Prob. Density 48
P(Time>48) = 0.04(55-48)
=0.28 or 28%
Lesson 1:
Continuous
Probability
Distribution

12. Unit 8: The Normal Distribution
Lesson 1: Continuous Probability Distributions
Lesson 1:
Continuous
Probability
Distribution