Question

Of all customers purchasing automatic garage door openers, 75% purchase chain-driven model. Let X = the number among the next 15 purchasers who select the chain-driven model. a. What is the frequency function (pmf) of X?

Answer 1

Answer:

         $P(X=x)=C(15,x)\cdot(0.75)^x\cdot(0.25)^{(15-x)}$

Explanation:

The probability mass function (PMF), or frequency function, is the function that gives the probabilities that a discrete random variable take some values.

In this problem, it is requested the frequency function (PMF) for the number of purchasers, among the next 15, who select a chain-driven model.

Then , you need to find, the function that gives P(X=0), P(X=1), P(X=2), P(X=3), . . . up to P(X=15).

Such as any function, the frequency  function can be presented as a formula, as a table, or as a graph.

Note that the statement represents a binomial disbribution in which success is that a customer select a chain-driven model and the fail is that a cusotmer does not select a chain-driven model.

The binomial probability for X = the number among the 15 purchasers who select the chain-driven model is given by the formula:

           $P(X=x)=C(n,x)\cdot(p)^x\cdot(1-p)^{(n-x)}$

Where:

•  $C(n,x)=\dfrac{n!}{x!(n-x)!}$

•   n is the number of times the experiment is performed: 15 in our problem

• p is the probability of succes: 0.75 in our problem

• 1-p is the probability of fail: 0.25 in our problem

Then, substitute:

           $P(X=x)=C(15,x)\cdot(0.75)^x\cdot(0.25)^{(15-x)}$

That is the frequency function.

If you want to give it as a table you must find P(X=1), P(X=2), P(X=3), . . . up to P(X=15) using that function. That is not part of the question.