Binomial Distribution PDF

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short lesson on binomial distribution...

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Course: Business Statistics Professor: Rocco Mosconi Title: The binomial distribution Agenda: • Binomial distribution • Binomial distribution in Excel • Expected value and variance of the Binomial distribution Binomial distribution This clip illustrates the Binomial probability distribution. The Binomial distribution is appropriate to represent the probabilities associated with a specific type of random variable: the sum of successes when we replicate a binomial experiment n times. As an economic example of reference, we could think of a car dealer who, in one day, proposes the purchase of a car to 10 potential buyers (n is equal to 10): how many cars will he be able to sell? We define by “X-i” the random variable "outcome of the i-th proposal".

the outcome of each proposal is random and binary: 1 if the sale is successful, 0 otherwise. We could therefore represent X-i as a Bernoulli random variable. Assume that the probability of success is the same with all potential customers, and denote it by π. We also assume that the n random variables are independent, i.e. that the outcome of the i-th proposal does not change the probability of success of the next (so, a customer is not influenced by seeing that the previous customer has bought or not). The sum of the outcomes can be written as follows:

Since the terms of the summation are random variables, the result is also a random variable. How is Y distributed? It can be shown that the probability distribution of Y, which is called the Binomial Distribution, is as follows

The formula may seem complex, but as we will soon see, this formula is available in a user friendly way in many programs, including excel: its practical use is therefore very easy. As we can see, the formula of the Binomial distribution is quite similar to that of the Bernoulli distribution. The binomial distribution has two parameters: π (real number between 0 and 1) and n (positive integer). Clearly, the variable Y can take n+1 values (in our example 11): 0, 1, 2, up to n The symbol "n over y" is called binomial coefficient. It is a function of n and y, and is needed here to ensure that sum of the probabilities is equal to 1. “n exclamation mark” reads " factorial n", and represents the product of all the integers from 1 to n. The hypotheses underlying this distribution are: - the n individual Bernoulli experiments have the same success probability - and they are independent Knowing π, the formula of “p of y” allows us to calculate, for example, the probability that the dealer sells 3 cars

First we need to calculate the value of the binomial coefficient. Once this is done, the calculation of the probability is simple. If, for example, the probability of selling a car to a generic customer is 10%, the probability of selling 3 cars when we meet 10 customers is 5.74%.

Binomial distribution in Excel The formula of “p of y” is clearly too complex to be calculated manually. Many programs make it easy to calculate using appropriate functions. Now let's see how the Binomial distribution is programmed in Excel:

In this sheet the first column represents the 11 possible values that our variable can assume, that is the number of cars sold by the dealer. The probability of success in the individual sale is in cell D1. To calculate the probability of a certain number of successes we can use an appropriate Excel function: we look for the function that represents the binomial distribution among the statistical functions. At this point we must introduce the number of successes, the number of attempts, the probability of success and, in this field called “cumulative”, we write 0. We see that the probability of 0 cars sold is equal to 34.8%; copying this formula throughout the column we find the probabilities of each possible outcome, it is easy to see that the sum of the probabilities is equal to 1. If we write 1 in the field “cumulative”, we get a slightly different result. I do exactly the same as before, but in the field “cumulative” I write 1, then I copy over the whole column. For example, this 73%, which corresponds to the probability of one car, indicates the sum of the probability of selling 0 cars plus the probability of selling 1 car, that is why we say “cumulative” probability; this 93% is equal to the sum of the first three probabilities. Therefore, it is the probability that the retailer sells less than 3 cars. For this probability distribution, we can also give a graphical representation this way: “insert”, “scatter”; the plot represents the probability corresponding to 0, 1, 2, 3 etc., all the possible outcomes, and we see that essentially it draws a curve. We note that if the probability of success in each single proposal is 10%, it is impossible to sell 10 cars out of 10 proposals. The most likely event is to sell 1 car, an event whose probability is 38.7%. The probability of not being able to sell any car is equal to 34.9%, while the probability of 2 successes is equal to 19.4%. The cumulative probability is read as follows: the probability of selling less than 4 cars (i.e. 0 or 1 or 2 or 3) is 98.7%. We notice how the shape of probability distribution changes when the parameter ฀฀ changes. If, for example, we change the probability of success from 10% to 50%, the shape of the curve becomes symmetric. If instead we consider 90%, the curve would display exactly the opposite asymmetry with respect to what we observed in the first case.

Expected value and variance of the Binomial distribution We can use excel to compute the expected value and the variance of the Binomial distribution. Let’s recall the formulae used to compute the expected value and the variance:

In Excel, as we have seen above, we can calculate the product of every possible value by the corresponding probability, and then make the sum of all these products; this gives the expected value.

Then we can calculate for every possible value the squared distance of that value with respect to the expected value, which we then multiply by the corresponding probability. Then we add this sum of products; this way, we get the variance. We can also calculate the standard deviation, which is nothing but the square root of the variance. We see that in this case, the expected number of cars that we expect the dealer to sell is equal to 1, with a variance of 0,9 and therefore a standard deviation of about 0.95. Again, changing the value of the probability, expected value, variance and standard deviation will change. For the Binomial distribution, it is not difficult to prove that the formulae can be simplified:

The expected value is equal to n times the probability of success, while the variance is equal to n times π times (1-π). It is easy to check in Excel that these formulae are correct....