1. 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

2. 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.

3. 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

4. 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) Experiment roll 2 die (fair or otherwise) X = Total score Y = Product of the scores Z = Difference in the scores

5. 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) Experiment roll 2 die (fair or otherwise) X = Total score Y = Product of the scores Z = 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

6. 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) Experiment roll 2 die (fair or otherwise) X = Total score Y = Product of the scores Z = 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

7. 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:

8. 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’? Outcome123456 Probabilityk2k3k4k5k6k

9. 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’? Outcome123456 Probability1/212/213/214/215/216/21 Total23456789101112 prob1/362/363/364/365/366/365/364/363/362/361/36 Total0123 prob1/83/83/361/8 Total012345 prob6/3610/368/366/364/362/36

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

11. If you roll a fair, six sided die lots of times and calculate the average score What answer would you expect to get? Outcome123456 Probability 1/61/6 1/61/6 1/61/6 1/61/6 1/61/6 1/61/6 Formula for real dataFormula using probabilities Often described as: mean of squares – square of mean Often described as: E(X 2 ) – (E(X)) 2

12. 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. Outcome123456 Probabilitykk2k3k0.23k

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