Question

Which one of the following may be the parameter of a binomial distribution?
A. \[np = 2,npq = 4\]
B. \[n = 4,p = \dfrac{3}{2}\]
C. \[n = 8,p = 1\]
D. \[np = 10,npq = 8\]

Answer 1

Hint: In the above given question, we are given four different options which are the probability of outcomes of the events \[p\] or \[q\] or both. We have to determine which one of the following outcomes can be the parameter of a binomial distribution. In order to approach the solution, first we have to recall the property of binomial distribution that the value of a probability of any outcome always lies between the values \[0\] and \[1\] .

Complete answer:
Given that, four different values for the outcomes of some events \[p\] and \[q\] .
We have to determine which one of them can be the parameter of binomial distribution.
We know that a probability of any outcomes in a binomial distribution lies between \[0\] and \[1\] .
Therefore, for any two events \[p\] and \[q\] , we must have the probabilities of \[p\] and \[q\] be as \[\;0 < p,q < 1\] such that \[p + q = 1\] to be the parameter of a binomial distribution.
Now we have to check if the four options satisfy this condition for a binomial distribution or not.
A. \[np = 2,npq = 4\]
Substituting the first value in the second equation, we have
\[ \Rightarrow \left( {np} \right)q = 4\]
That gives us,
\[ \Rightarrow 2 \cdot q = 4\]
That is,
\[ \Rightarrow q = 2\]
Since, \[q > 1\] hence it can not be a parameter of a binomial distribution.

B. \[n = 4,p = \dfrac{3}{2}\]
Here, we have \[p = \dfrac{3}{2} = 1.5 > 1\] , hence it is also incorrect.

C. \[n = 8,p = 1\]
Since \[p = 1\] instead of \[p < 1\] , hence it is also an incorrect option.

D. \[np = 10,npq = 8\]
Substituting the first value in the second equation, we have
\[ \Rightarrow \left( {np} \right)q = 8\]
That gives us,
\[ \Rightarrow 10 \cdot q = 8\]
That is,
\[ \Rightarrow q = 0.8\]
Now since, \[q < 1\] therefore, it can be a parameter for a binomial distribution.
Therefore, the correct option is D

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