If the mean of the binomial distribution is 100. Then standard deviation lies in the interval

Options

A. [0, 7)
B. [1, 7)
C. [0, 10)
D. [1, 11)

Answer

To solve the problem, we need to find the interval in which the standard deviation of a binomial distribution lies when the mean is given as 100.

### Step-by-Step Solution:

1. **Understand the Mean of Binomial Distribution**:
The mean (μ) of a binomial distribution is given by the formula:
\[ \mu = n \cdot p \]
where \( n \) is the number of trials and \( p \) is the probability of success. Given that the mean is 100, we have:
\[ n \cdot p = 100 \]

2. **Understand the Standard Deviation of Binomial Distribution**:
The standard deviation (σ) of a binomial distribution is given by the formula:
\[ \sigma = \sqrt{n \cdot p \cdot q} \]
where \( q = 1 - p \) (the probability of failure).

3. **Set Up the Inequalities**:
Since \( p \) (the probability of success) must lie between 0 and 1, we have:
\[ 0 < p < 1 \]
Consequently, \( q \) must also lie between 0 and 1:
\[ 0 < q < 1 \]

4. **Express \( q \) in Terms of \( p \)**:
We know that:
\[ q = 1 - p \]
Therefore, we can express the product \( p \cdot q \) as:
\[ p \cdot q = p \cdot (1 - p) = p - p^2 \]

5. **Find the Maximum Value of \( n \cdot p \cdot q \)**:
Since \( n \cdot p = 100 \), we can substitute this into the standard deviation formula:
\[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{100 \cdot q} \]
Now, we need to find the maximum value of \( q \) given \( p \) is between 0 and 1.

6. **Determine the Range of \( q \)**:
Since \( p \) varies from 0 to 1, \( q \) will vary from 1 to 0. The maximum value of \( q \) occurs when \( p \) is at its minimum (close to 0), which means \( q \) approaches 1.

7. **Calculate the Standard Deviation**:
Therefore, we have:
\[ \sigma = \sqrt{100 \cdot q} \]
Since \( q \) can be at most 1, we have:
\[ \sigma \leq \sqrt{100} = 10 \]
And since \( p \) cannot be 0 (otherwise the mean would not be defined), \( q \) cannot be 1. Hence, \( q \) must be greater than 0, which implies:
\[ \sigma > 0 \]

8. **Conclusion**:
Thus, the standard deviation lies in the interval:
\[ 0 < \sigma < 10 \]
Therefore, the correct option is that the standard deviation lies in the interval \( [0, 10) \).