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S1 Chapter 6: The Binomial Distribution Using the Binomial Distribution
Mr J Wooler firstname.lastname@example.org
In Chapter 4, we looked at Discrete Random Variables in general.
We looked at methods of calculating the expected value and the standard deviation.
We are now going to find out how to calculate the expected value of a random variable modelled by the Binomial Distribution.
The Expected Value
The expectation of is given by
This seems obvious, since if the probability of success in each independent trial is , then the expected number of successes in trials is .
Let’s try and prove it though.
First consider . The probability distribution looks like this:
Since , .
The general case follows the same pattern, with the common factor being and the expectation simplifying to
Find the expectation of X
What is the most likely outcome for X ?
The most likely outcome for X is the value of X with the highest probability.
The most likely outcome is .
Example 2 – Exam-Style Question
Very careful reading is required in this question. In parts (iii) and (iv) you are using a previous answer as the probability in a new binomial situation.
Finding an unknown sample size
Estimating the probability from experimental data
Using the Binomial Distribution
Pages 158 – 162
Questions 1, 2, 3, 4, 8, 9, 11 & 12
Past Exam Question