1. M248: Analyzing data
Block A
UNIT A3
Modeling Variation

2. UNIT A3: Modeling variation
Block A
UNIT A3: Modeling variation
Contents
Section 1: What is probability?
Section 2: Modeling random variables
Section 3: Describing probability distributions
Section 6: The Normal Distribution
Terms to know and use
Unit A3 Exercises

3. Section 1: What is probability?
Probability is a number between 0 and 1 inclusive which measures how likely (chance) an event is to occur (happen).
There are two approaches to probability:
Finding probability from a constructed model.
Example: The experiment of a perfect die or a fair coin.
Estimating a probability from data (Relative Frequency)
Example:
Not Satisfied
Satisfied
Highly Satisfied
Total 40 35 25
100

4. An event (E) consists of a number of outcomes.
Section 1: What is probability? Finding probability from a constructed model
A probability experiment is a chance process that leads to well-defined results called outcomes.
An outcome is the result of a single trial of a probability experiment.
A sample space is the set of all possible outcomes of a probability experiment.
An event (E) consists of a number of outcomes.

5. Experiment Sample Space (S) Toss a coin Head, Tail Roll a die
Section 1: What is probability?
Finding probability from a constructed model
Experiment
Sample Space (S)
Toss a coin
Head,
Tail
Roll a die
1, 2, 3, 4, 5, 6
Answer a true/false
True, False
question
Tossing a coin twice
HH,
HT, TH, TT

6. Section 1: What is probability
Section 1: What is probability? Finding probability from a constructed model
To find a probability value we use sample spaces
All outcomes in the sample space are equally likely to happen.
Experiment
Sample Space
Probability
Rolling a die
{1,2,3,4,5,6}
P(even)=3/6=0.5
Having 2 Babies
{BB,BG,GB,GG}
P(2 boys)=1/4
Answering a True or False question
{T,F}
P(Correct answer)=1/2
P(E) is usually read as ‘probability of E’ or simply as ‘P of E’

7. Section 1: What is probability
Section 1: What is probability? Finding probability from a constructed model

8. Section 1: What is probability
Section 1: What is probability? Finding probability from a constructed model
If the probability that a person lives in an industrialized country of the world is , find the probability that a person does not live in an industrialized country.

9. Section 1: What is probability?
Estimating probability from data
Estimating probability from data depends on actual experience to determine the likelihood of outcomes.
The probability that an event E occurs is the proportion of times that E occurs. It is the sample relative frequency of E.

10. Section 1: What is probability?
Estimating probability from data
In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities.
a. A person has type O blood.
Type
Frequency A 22 B 5 AB 2 O 21
Total 50

11. Section 1: What is probability?
Estimating probability from data
In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities.
b. A person has type A or type B blood.
Type
Frequency A 22 B 5 AB 2 O 21
Total 50

12. Section 1: What is probability?
Estimating probability from data
In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities.
c. A person has neither type A nor type O blood.
Type
Frequency A 22 B 5 AB 2 O 21
Total 50

13. Section 1: What is probability?
Estimating probability from data
In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities.
d. A person does not have type AB blood.
Type
Frequency A 22 B 5 AB 2 O 21
Total 50

14. Section 1: What is probability? Properties
Properties of probability:
For any event E, we have
0 ≤ P(E) ≤ 1
If an event E is impossible, then P(E) = 0
If an event E is certain to happen, then P(E) = 1
P(E) cannot be negative
Note: you have to read and solve all the activities and exercises in this section

15. Section 2: Modeling random variables What is a random variable?
A random variable is a variable whose possible values are numerical assigned to all possible outcomes of an experiment. For example, the result of rolling a die.
Random variables take numeric values. The set of values that a random variable can take is called the range of the random variable.
Example: the range of the random variable X assigned to the value recording when rolling a die is: {1,2,3,4,5,6}

16. Section 2: Modeling random variables Section 2
Section 2: Modeling random variables Section 2.1: Discrete and continuous random variables
A random variable is said to be a discrete random variable when the variable takes only a discrete set of values. Discrete random variables are usually counts.
Example: Household size (number of family members), rolling a die.
A random variable is said to be a continuous random variable when the variable can take any value within a continuous range of values. Continuous random variables are usually measurements.
Example: Weight of a newborn baby.
Since random variables take numeric values, then when we have a categorical variable we should associate a number for each event, and then define the random variable.
Example: Tossing a coin, 0 assigned for Head and 1 for Tail.

17. Section 2: Modeling random variables Section 3: Describing probability distributions Discrete random variable (Probability distribution and Probability Mass Function)
Example: If we take a six-sided die and roll it n times. Each time the die shows a three or a six-spot face, you record a 1. If it shows a different number (1,2,4 or 5) you record a 0. If we denote the result recorded after a single roll of the die by X, Then X can take one of the two values 0 and 1. In this case we have:
X is a discrete random variable
Range of X is {0,1}

18. Section 2: Modeling random variables Section 3: Describing probability distributions Discrete random variable (Probability distribution and Probability Mass Function)
Now assuming that the die is perfect, the probability that X takes the value 1 is 2/6 ( 2 out of 6 possible outcome) . The probability that X takes the value 0 is 4/6. Using the notation of probability:
P(X = 1) = 1/3, P(X = 0) = 2/3 (Probability Distribution)
A Probability Distribution table will summarize the last 2 slides:
(Probability Mass Function) x 1
P(X=x)
2/3
1/3

19. Diagram of the theoretical probability distribution
Section 2: Modeling random variables Section 3: Describing probability distributions Discrete random variable (Probability distribution and Probability Mass Function)
Diagram of the theoretical probability distribution
Properties of probability mass function
For a discrete random variable X with p.m.f p(x) we have:
and

20. F(x) = The sum of all probabilities less than or equal to x
Section 3: Describing probability distributions Discrete random variable (The Cumulative distribution function, c.d.f)
The cumulative distribution function F of a random variable X is a function which, for each value x in the range of X, gives the probability that X takes a value less than or equal to x:
F(x) = The sum of all probabilities less than or equal to x
Example: F(1)= P(X=1) + P(X=0)
F(3) = P(X=3) + P(X=2) + P(X=1) + P(X=0)+….
Note: The notation F(.) is standard for a c.d.f, for both discrete and continuous random variables
Solve activity 3.1 page 101

21. Weight of a newborn baby Frequency (# of babies)
Section 2: Modeling random variables Section 3: Describing probability distributions Continuous random variable (Probability distribution and Probability Density Function)
Consider the example below
Weight of a newborn baby
Frequency (# of babies)
Relative Frequency 3
0.06 5
0.1 7
0.14 12
0.24 9
0.18 4
0.08
3.6-4
Total 50 1

22. Section 2: Modeling random variables Section 3: Describing probability distributions Continuous random variable (Probability distribution and Probability Density Function)
The function which defines the equation of such a distribution is called the probability density function (p.d.f).
Note: Determining the p.d.f for the above distribution is not an easy task

23. Section 2: Modeling random variables Section 3: Describing probability distributions Continuous random variable (Probability Density Function)
The probability density function (p.d.f)
The probability function for a continuous random variable is usually called the probability density function of the random variable. A p.d.f f(x) defines a curve.
Properties of probability density function
The p.d.f is always positive.
Area under the curve equals to 1.

24. Section 3: Describing probability distributions Continuous random variable (The Cumulative distribution function c.d.f)
The cumulative distribution function F of a random variable X is a function which, for each value x in the range of X, gives the probability that X takes a value less than or equal to x:
For each value x in the range of X, F(x) is equal to the area under the graph of the p.d.f to the left of x.
Solve activity 3.2 page 101 to 103

25. Section 6: The normal distribution
When the shape of the histogram of a continuous model suggests that the graph of the p.d.f has a single peak and is symmetrical about this peak, Then this model is called the normal distribution or the Gaussian distribution.
The probability density function of a typical normal distribution is given by:
where are two constant parameters of the normal distribution,
X has a normal distribution with parameters This is written

26. Section 6: The normal distribution
The p.d.f of the normal family is symmetric about Observations less than about or more than are rather unlikely

27. Terms to know and use Probability Random variable
Probability distribution
Probability mass function (p.m.f)
Probability density function (p.d.f)
Cumulative distribution function (c.d.f)

28. Unit A3 Exercises M248 Exercise Booklet
Solve the following exercises:
Exercise 11 ………………………………………. Page 6
Exercise 12 ………………………………………. Page 7
Exercise 13 ………………………………………..Page 7
Exercise 14 ………………………………………..Page 7
Exercise 15 ………………………………………..Page 7