Question

Select a discrete probability distribution and present a real-life application of that distribution. Interpret the expected value and the standard deviation of your selected distribution within the context of the real-life example that you have selected, and describe how these values can be used by enterprise decision-makers.

Answer 1

If a new product wants to be tested by a company and decides to show 50 samples of this product to 50 selected customers. The company estimates that the probability that the customer buys the product is 0.67, the objective is to determine approximately how many people expect to buy the product.

Let X the random variable of interest "Number of people that will buy a selected product", on this case we now that:

`X \sim Binom(n=50, p=0.67)`

The expected value is given by this formula:

`E(X) = np=50*0.67=33.50`

And the standard deviation for the random variable is given by:

`sd(X)=√(np(1-p))=√(50*0.67*(1-0.67))=3.32`

So then they can conclude that for each group of 50 people they expect that about 33-34 peoploe will buy the product with a standard deviation of 3.32.

Explanation:

Previous concepts

A Bernoulli trial is "a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted". And this experiment is a particular case of the binomial experiment.

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

The probability mass function for the Binomial distribution is given as:

`P(X)=(nCx)(p)^x (1-p)^(n-x)`

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`

Solution to the problem

If a new product wants to be tested by a company and decides to show 50 samples of this product to 50 selected customers. The company estimates that the probability that the customer buys the product is 0.67, the objective is to determine approximately how many people expect to buy the product.

Let X the random variable of interest "Number of people that will buy a selected product", on this case we now that:

`X \sim Binom(n=50, p=0.67)`

The expected value is given by this formula:

`E(X) = np=50*0.67=33.50`

And the standard deviation for the random variable is given by:

`sd(X)=√(np(1-p))=√(50*0.67*(1-0.67))=3.32`

So then they can conclude that for each group of 50 people they expect that about 33-34 peoploe will buy the product with a standard deviation of 3.32.