Which one of the following may be the parameter of a binomial distribution?

Options

A. A. `np=2, npq= 4`
B. B. `n=4, p = (3)/(2)`
C. C. `n= 8, p=1`
D. D. `np= 10, npq= 8`

Answer

To determine which of the given options may be the parameter of a binomial distribution, we need to analyze each option based on the properties of binomial distributions.

### Step-by-Step Solution:

1. **Understand Binomial Distribution Parameters**:
- A binomial distribution is defined by two parameters: \( n \) (the number of trials) and \( p \) (the probability of success in each trial).
- The probabilities \( p \) and \( q \) (where \( q = 1 - p \)) must satisfy \( p + q = 1 \).
- Both \( p \) and \( q \) must be in the range [0, 1].

2. **Evaluate Each Option**:
- We will evaluate each of the four options provided to see if they satisfy the conditions of a binomial distribution.

3. **Option A**: \( np = 2 \) and \( npq = 4 \)
- From \( np = 2 \), we have \( n \cdot p = 2 \).
- From \( npq = 4 \), substituting \( np \) gives \( 2q = 4 \) which simplifies to \( q = 2 \).
- Since \( q = 2 \) is not in the range [0, 1], this option is invalid.

4. **Option B**: \( n = 4 \) and \( p = \frac{3}{2} \)
- Here, \( p = \frac{3}{2} \) is greater than 1, which is not possible for a probability.
- Therefore, this option is also invalid.

5. **Option C**: \( n = 8 \) and \( p = 1 \)
- If \( p = 1 \), then \( q = 1 - p = 0 \).
- While \( p \) is valid, having \( q = 0 \) means there are no failures, which is technically valid but not a typical scenario in binomial distributions.
- However, since \( q \) can be zero, this option is valid.

6. **Option D**: \( np = 10 \) and \( npq = 8 \)
- From \( np = 10 \), we have \( n \cdot p = 10 \).
- Substituting into \( npq = 8 \) gives \( 10q = 8 \) which simplifies to \( q = \frac{8}{10} = \frac{4}{5} \).
- Then, \( p = 1 - q = 1 - \frac{4}{5} = \frac{1}{5} \).
- Both \( p \) and \( q \) are valid probabilities (in the range [0, 1]), and we can find \( n \) as \( n = \frac{10}{p} = \frac{10}{\frac{1}{5}} = 50 \).
- This option is valid.

### Conclusion:
- The valid options are C and D, but since we are looking for a single answer, the most typical parameter set for a binomial distribution is found in **Option D**.

### Final Answer:
**Option D**: \( np = 10 \) and \( npq = 8 \) is a valid parameter of a binomial distribution.