BINOMIAL DISTRIBUTION
Applied Statistics and Computing Lab
Indian School of Business
Learning goals •
To understand idea behind Binomial distribution
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To study the behaviour of Binomially distributed variables
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To look at examples of Binomial distribution
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To understand the importance of the t he distribution
Price increases Price increases
Day 4
Price decreases or remains the same Price increases
Day 3 Price increases Price decreases or remains the same
Price of stock A
Day 2
Day 4
Price decreases or remains the same ………… and so on Price increases
Day 1 Price increases
Price decreases or remains the same
Day 4
Price decreases or remains the same
Day 3 Price increases Price decreases or remains the
Day 4
Example •
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Suppose we know the values of and for stock A and they remain the same every day We can easily see that + = 1 Can we find the probability that ‘during the 5 trading days of next week, the price of stock A will increase on 3 of the days?’ This is an experiment with only two possible outcomes viz. ‘the price will increase’ or ‘the price will decrease or remain the same’ The probability of either of these possibilities remains same on each day i.e. in each trial Let us assume that whether the price increases, decreases or remains the same on o n a particular day, day, does not have any influence on whether the price increases, decreases or remains the same on the next day
Calculating the probability probability •
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Suppose that event = price increases on (Day 1, day 2, day 3) and decreases or remains the same on (Day 4, day 5) By using rules of probability P = ( × )
where, = ℎ = ℎ ℎ •
In how many ways can the price increase on 3 out of 5 days?
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Using the counting principle,
. ℎ ℎ 3 5 =
5 3
Bringing together all possibilities Here are 6 mutually exclusive and collectively exhaustive events events under this experiment:
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= price increases on exactly 0 out of 5 days
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= price increase on exactly 1 out of 5 days
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= price increase on exactly 2 out of 5 days
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= price increase on exactly 3 out of 5 days
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= price increase on exactly 4 out of 5 days
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6 = price increase on exactly 5 out of 5 days
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Consider . Each simple event contributing to has probability ( × − )
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Further, there are
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What if we sum all the possibilities and the corresponding corresponding probabilities? 5 0
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( × − )+
5 1
simple events contributing to
( × − )+
5 2
( × − )+
5 3
( × − )+
5 4
( × − )+
5 5
( × − )
We can check that this adds up to 1 as they are probabilities of 6 mutually exclusive and
Name of the distribution distribution •
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The earlier summation of probabilities, probabilities, is simply the binomial expansion of ( + ) In general, suppose is the probability of achieving one of the two mutually exclusive and collectively exhaustive outcomes of an event We term = (1 ) and consider the binomial expansion of ( + ) For k = 0,1,2, … , , every term of this expansion is of the form (−) In the binomial expansion,
is the binomial coefficient of
A variable with such a probability probability distribution is called the Binomial distribution!
Binomial distribution •
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The stock prices example discussed above is an example example of Binomial distribution A binomial distribution is characterised by, by, –
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An experiment experiment that can result in only one of o f the two possible outcomes, one of which is termed to be success and other as failure Constant probability of ‘success’ at every trial An experiment experiment consisting of multiple trials, none of which influences the outcome of any other trial (independent trials)
The probability of success is generally termed and therefore the probability of failure is ( 1 ) The stock price experiment had each of these features! features!
Binomial distribution (contd.) (contd.) •
The parameters parameters of a binomially distributed distributed variable are ‘Number of trials’ () and ‘Probability of success’ and we denote it as ~(, ~ (, )
The probability mass function (PMF) for such an is given by, = = () − where, = 0,1,2, … , + = 1 and 0 ≤ ≤ 1 •
Bernoulli trial • •
What if our interest is in only one trial? Bernoulli trial or Bernoulli distribution is characterised by –
An experiment that can result in only one of the two possible outcomes, one of which is termed to be success (with probability ) and other as failure (with probability ( 1 ))
For a Bernoulli random variable , we denote ~() = = − where, = 0,1 and + = 1 and 0 < < 1 •
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For this , E = and V = (1 ) Binomial distribution is just an extension extension of Bernoulli trials, to independently and identically distributed Bernoulli random variables
Scope of Binomial distribution di stribution •
More situations where binomial distribution can help us evaluate probabilites of certain events: –
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If an insurance company knows the probability of a claim being fraudulent (or not), binomial distribution can help determine the probability that there would be more than ‘k’ fraudulent claims in the next 1000 claims If an advertising company knows the probability of a customer buying a certain product, if he/she receives an advertisement on their Facebook account, binomial distribution can help them determine whether ‘k’ purchases would be made from a total of 10000 Facebook ads sent. Effectively, they can determine how many Facebook advertisements to send, if they want to achieve a particular sales target Suppose that a hardware store manager has 3 different different suppliers. If he knows the probability of a newly purchased bolt being defective, based on the supplier it is bought from, binomial distribution can help him/her check whether there will be less than ‘k’ defective bolts in a box of 2000 bolts. This would help his decide which supplier to buy the bolts from. We know that the probability of a newly born child being a boy is equal to the probability that it is a girl, 0.5. Binomial distribution can help us determine the
Binomial distribution (contd.) (contd.) •
Can coin tossing be considered as an example of binomial distribution?
1. A coi coin n toss toss can can onl only y res resul ultt in in a head head or a tail ail 2. The The prob probab abil ilit ity y of obs observ ervin ing g a head head,, is con const stan antt at each toss 3. The The out outcome come of of one one tos tosss (whe (wheth ther er head head or or tai tail) l) doe doess not influence the outcome of the next toss •
Any Bernoulli variable can be modelled based on a coin tossing experiment!
History Independent trials having a common success probability ‘p’ were first studied by the Swiss mathematician Jacques Bernoulli (1654-1705). In his book Ars Conjectandi (the art of conjecturing), published by his nephew Nicholas eight years after his death in 1713, Bernoulli Berno ulli showed that if the number of such trials were large, then the proportion of them that were successes would be close to p.
Alternate Alternate definition of success and failure • • •
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In a Bernoulli trial, the definition definition of success and failure is relative A success or a failure can contain multiple simple events ‘ The share price will decrease’ or ‘the share price will remain constant’ constant ’ are two simple events In our example, we combined these into one outcome This combined outcome, along with the possible outcome ‘the share price will increase’ give two mutually exclusive and collectively exhaustive events with respect to what may happen to the price of the share Therefore we could consider it to be a Bernoulli trial Similarly, we could look at something as simple as roll of a die It has six possible outcomes, but if we define events as ‘even number’ or ‘odd number’, we can find the probabilities using Binomial distributio
Properties Properties of Binomial distribution •
For a ~(, ~( , )
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V =
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Suppose we want to calculate the probability that in the next 120 trading days, the price of stock A will increase 37 times 120 = 37 = − 37 Very difficult to calculate! As the value of becomes very large, it becomes increasingly difficult to calculate the probabilities of Binomial distribution Properties of Binomial distribution help us approximate these probabilities probabilities easily
Properties of Binomial distribution (contd.) As value of increases
If is very large, is very small and is finite, Binomial distribution tends to Poisson distribution
Poisson distribution allows to calculate the probability of ‘’ successes without specifying the number of
As becomes very large, Binomial distribution tends to Normal distribution
This is a very important property and we will learn the significance of Normal distribution at a later time
Thank you
