Lesson ObjectiveUnderstand what we mean by a Random Variable in mathsUnderstand what is meant by the expectation and variance of a random variableBe able to calculate the Expectation and Variance of a discrete random variable

Suppose you conduct an experiment where the outcomes are unpredictableSuppose you assign a number to each of the outcomes.

Then the way that you assign numbers to the outcomes is called a Random VariableWe tend to use capital letters to define Random Variables.

Suppose you conduct an experiment where the outcomes are unpredictableSuppose you assign a number to each of the outcomes.

Then the way that you assign numbers to the outcomes is called a Random VariableWe 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 dieY = The number of times that the die bouncesZ = The time it takes for the die to stop movingW = The score on the die divide by 2V = The number of factors in the number that you scored

Suppose you conduct an experiment where the outcomes are unpredictableSuppose you assign a number to each of the outcomes.

Then the way that you assign numbers to the outcomes is called a Random VariableWe 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 dieY = The number of times that the die bouncesZ = The time it takes for the die to stop movingW = The score on the die divide by 2V = The number of factors in the number that you scored

2) Experiment roll 2 die (fair or otherwise)X = Total scoreY = Product of the scoresZ = Difference in the scores

Suppose you conduct an experiment where the outcomes are unpredictableSuppose you assign a number to each of the outcomes.

Then the way that you assign numbers to the outcomes is called a Random VariableWe 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 dieY = The number of times that the die bouncesZ = The time it takes for the die to stop movingW = The score on the die divide by 2V = The number of factors in the number that you scored

2) Experiment roll 2 die (fair or otherwise)X = Total scoreY = Product of the scoresZ = 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 dieY = Number of Heads minus the score on the die score

Suppose you conduct an experiment where the outcomes are unpredictableSuppose you assign a number to each of the outcomes.

Then the way that you assign numbers to the outcomes is called a Random VariableWe 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 dieY = The number of times that the die bouncesZ = The time it takes for the die to stop movingW = The score on the die divide by 2V = The number of factors in the number that you scored

2) Experiment roll 2 die (fair or otherwise)X = Total scoreY = Product of the scoresZ = 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 dieY = Number of Heads minus the score on the die score

4) Experiment: Weigh someoneX = 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

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 1/21 2/21 3/21 4/21 5/21 6/21

Total 2 3 4 5 6 7 8 9 10 11 12

prob 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

Total 0 1 2 3

prob 1/8 3/8 3/36 1/8

Total 0 1 2 3 4 5

prob 6/36 10/36 8/36 6/36 4/36 2/36

If you roll a fair, six sided die lots of times and calculate the average scoreWhat answer would you expect to get?

If you roll a fair, six sided die lots of times and calculate the average scoreWhat answer would you expect to get?

Outcome 1 2 3 4 5 6Probability 1/6

1/61/6

1/61/6

1/6

Expectation = E(X) = μ = Mean =

Root Mean Squared =

Variance = Variance = σ2 =

Formula for real data Formula using probabilities

Often described as: mean of squares – square of mean

Often described as: E(X2) – (E(X))2

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

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. Find the expected amount drawn from the bag and the variance.

Outcome 1 2 3 4 5 6

Probability k k 2k 3k 0.2 3k

1)

3) 4)

1)