Lesson Objective
Understand what we mean by a Random Variable in maths
Understand what is meant by the expectation and variance of a random variable
Be able to calculate the Expectation and Variance of a discrete random variable
Suppose you conduct an experiment where the outcomes are unpredictable
Suppose you assign a number to each of the outcomes.
Then the way that you assign numbers to the outcomes is called a Random Variable
We tend to use capital letters to define Random Variables.
Suppose you conduct an experiment where the outcomes are unpredictable
Suppose you assign a number to each of the outcomes.
Then the way that you assign numbers to the outcomes is called a Random Variable
We tend to use capital letters to define Random Variables.
Examples of Random Variables:
1) Experiment: Roll a die (fair or otherwise)
X = Score on the die
Y = The number of times that the die bounces
Z = The time it takes for the die to stop moving
W = The score on the die divide by 2
V = The number of factors in the number that you scored
Suppose you conduct an experiment where the outcomes are unpredictable
Suppose you assign a number to each of the outcomes.
Then the way that you assign numbers to the outcomes is called a Random Variable
We tend to use capital letters to define Random Variables.
Examples of Random Variables:
1) Experiment: Roll a die (fair or otherwise)
X = Score on the die
Y = The number of times that the die bounces
Z = The time it takes for the die to stop moving
W = The score on the die divide by 2
V = The number of factors in the number that you scored
2)
X
Y
Z
Experiment roll 2 die (fair or otherwise)
= Total score
= Product of the scores
= Difference in the scores
Suppose you conduct an experiment where the outcomes are unpredictable
Suppose you assign a number to each of the outcomes.
Then the way that you assign numbers to the outcomes is called a Random Variable
We tend to use capital letters to define Random Variables.
Examples of Random Variables:
1) Experiment: Roll a die (fair or otherwise)
X = Score on the die
Y = The number of times that the die bounces
Z = The time it takes for the die to stop moving
W = The score on the die divide by 2
V = The number of factors in the number that you scored
2)
X
Y
Z
Experiment roll 2 die (fair or otherwise)
= Total score
= Product of the scores
= Difference in the scores
3) Experiment: Roll a die and toss two coins (fair or otherwise)
X = Number of Heads plus the score on the die
Y = Number of Heads minus the score on the die score
Suppose you conduct an experiment where the outcomes are unpredictable
Suppose you assign a number to each of the outcomes.
Then the way that you assign numbers to the outcomes is called a Random Variable
We tend to use capital letters to define Random Variables.
Examples of Random Variables:
1) Experiment: Roll a die (fair or otherwise)
X = Score on the die
Y = The number of times that the die bounces
Z = The time it takes for the die to stop moving
W = The score on the die divide by 2
V = The number of factors in the number that you scored
2)
X
Y
Z
Experiment roll 2 die (fair or otherwise)
= Total score
= Product of the scores
= Difference in the scores
3) Experiment: Roll a die and toss two coins (fair or otherwise)
X = Number of Heads plus the score on the die
Y = Number of Heads minus the score on the die score
4) Experiment: Weigh someone
X = How heavy the person is
Random variables can be both discrete or continuous
We are only interested in discrete Random Variables for S1
A list of the outcomes of a random variable with their associated probabilities is called a
Distribution
Let X = Score when you roll a fair die
The distribution for X looks like this:
Draw distribution tables for the following:
1) Y = Total score when you roll 2 dice
2) X = Difference in the score when you roll 2 dice
3) Z = Number of Heads when you toss 3 coins
4) Consider the following distribution table.
What is the value of ‘k’?
Outcome
1
2
3
4
5
6
Probability
k
2k
3k
4k
5k
6k
Draw distribution tables for the following:
1) Y = Total score when you roll 2 dice
Total 2
3
4
5
6
prob
1/36
2/36
3/36
4/36
5/36
7
8
9
10
11
12
6/36
5/36
4/36
3/36
2/36
1/36
2) X = Difference in the score when you roll 2 dice
Total
0
1
2
3
4
5
prob
6/36
10/36
8/36
6/36
4/36
2/36
3) Z = Number of Heads when you toss 3 coins
Total 0
1
2
3
prob
3/8
3/36
1/8
1/8
4) Consider the following distribution table.
What is the value of ‘k’?
Outcome
1
2
3
4
5
6
Probability
1/21
2/21
3/21
4/21
5/21
6/21
If you roll a fair, six sided die lots of times and calculate the average score
What answer would you expect to get?
If you roll a fair, six sided die lots of times and calculate the average score
What answer would you expect to get?
Outcome
1
2
3
4
5
6
Probability
1/
6
1/
6
1/
6
1/
6
1/
6
1/
6
Formula for real data
Mean =
Formula using probabilities
𝑥×𝑓
𝑓
Expectation = E(X) = μ =
Root Mean Squared =
Variance =
𝑥 2 ×𝑓
𝑓
𝑥 2 ×𝑓
𝑓
𝑥 × 𝑝(𝑥)
− (mean)2
− (mean)2
Often described as:
mean of squares – square of mean
Variance = σ2 = 𝑥 2 × 𝑝(𝑥) − (expectation)2
Often described as:
E(X2) – (E(X))2
LESSON OBJECTIVE
Be able to find the expectation (theoretical mean) and theoretical
variance of a discrete probability distribution
Experiment:
Toss three coins
Random Variable: X = Difference in the number of heads and tails
a) Draw a probability distribution table for X
b) Calculate the expectation and variance of X
LESSON OBJECTIVE
Be able to find the expectation (theoretical mean) and theoretical
variance of a discrete probability distribution
Experiment:
Toss three coins
Random Variable: X = Difference in the number of heads and tails
a) Draw a probability distribution table for X
b) Calculate the expectation and variance of X
Out
Prob
xp
x2p
1
0.25
0.25
0.25
3
0.75
2.25
6.75
HHH
HHT
HTH
THH
TTH
THT
HTT
TTT
1
2.5EXP
0.75VAR
Diff 3
Diff 1
Diff 1
Diff 1
Diff 1
Diff 1
Diff 1
Diff 3
1) A 4 sided spinner labelled 1, 2, 3 and 4 is spun twice and the scores added together.
Draw a probability distribution table
Calculate the expected total score and the variance in the total score.
2) A probability distribution for a random variable Y is defined as shown
Outcome
1
2
3
4
5
6
Probability
k
k
2k
3k
0.2
3k
Calculate the Expectation and variance of Y
3)
4) A bag contains four Russian banknotes, worth 5, 10, 20 and 50 roubles respectively.
An experiment consists of repeatedly taking a note from the bag at random and
replacing it, before redrawing another note.
Find the expected amount drawn from the bag and the variance.
1)
2)
3)
Out
Prob
1
0.08
0.08
0.08
2
0.08
0.16
0.32
3
0.16
0.48
1.44
4
0.24
0.96
3.84
5
0.2
1
5
6
0.24
1.44
8.64
1
4.12 EXP
2.3456 VAR
4)