Aims: To know and apply the rules for expectation and variance

Mean, Variance and Standard Deviation of Binomial DistributionsAims: To know and apply the rules for expectation and variance.

Outcomes

Name: Know what is meant by Mean/Expected Value, Variance and in terms of Binomial Distributions.Describe: How to calculate the mean and variance (and therefore standard deviation) of a Binomial variable.Explain: How to justify/criticise the use of a Binomial Model.

Maths

p84 Ex3ENext Lesson: Normal DistributionRonald Fisher 1890-1962: Anders Hald called him "a genius who almost single-handedly created the foundations for modern statistical science", while Richard Dawkins named him "the greatest biologist since Darwin”.

Mean/Expected Value

Practical and easy to see example...If you roll a die what is P(4)?

If you rolled it 6 times how many times do you expect to get a 4?12 times18 times24 times100 times

For X~B(n,p)Mean = np

Variance

The Variance is a bit trickier.The formula for the variance is...

Example

Sam buys football cards; the chance he gets a foil card in a pack is 0.4.

Sam buys packs every day for 4 weeks.

What are the mean and standard deviation of the number of foil cards gained?

Appropriate Model?

Sometimes you will be asked to compare the mean and standard deviation of observed data to those of a Binomial Distribution to decide if a Binomial Distribution model is appropriate.

Possible IssuesYou may also be asked to suggest possible issues with a binomial (times that the conditions might not be satisfied).Usually this is why the data might not be independent (why the probability might change). Mean is same but SD/Var different.My favourite was why presence of pensioners on a bus might not be independent from day to day!You have to use a little sense (or not as the case may be).