Binomial distribution
Random variables are a type of variable whose value depends on the outcome of a random
event.
The sample space of a random variable is the range of values the variable can assume.
The below terms can be used to describe random variables:
Random โ Probability not known until the experiment is carried out.
Discrete โ The variable can only take certain values e.g. in the sample space
๐ฅ=1, 2, 3, 4
The probability distribution can be denoted by the probability mass function for a set of
outcomes, i.e. the sample space - this can be done using a table, graph or standard written
notation for DRV.
The probability that a random variable, , takes a particular value is written as
๐ ๐ฅ ๐(๐=๐ฅ)
{for all } *
ฮฃ๐(๐=๐ฅ)=1 ๐ฅ
You should know the conditions to model with a binomial distribution
๐ ๐ต(๐,๐)
For instance, imagine you are given that the probability mass function is
๐ ~ ๐ต(12,1
6)
We know that there is a ๏ฌxed number of trials: trials.
12
The probability of success is ; let this be the probability of rolling a on a dice.
1
66
If we want to ๏ฌnd the probability that we roll a a total of three times, we can sayโฆ
6
๐(๐=3)=ยนยฒ๐ถโร(1
6)ยณร(5
6)โน
*Remember that for DRVs if the probability distribution is not uniform we can use this fact
to help.
E.g. spinner is spun a maximum of four times; the trial ends when the spinner lands on red
For binomial PD, is the probability of an event occurring with a ๏ฌxed probability.
๐
For binomial CD, we take the cumulative probability function, which tells us the sum of all
the individual probabilities up to and including the given value in the calculation for
๐ฅ
๐(๐=๐ฅ)
IMPORTANT QUESTIONS
