If for a poisson distribution P(X=0)=0.2 then the variance of the distribution is

Options

A. 2
B. 1
C. e
D. `log5`

Answer

To solve the problem, we need to find the variance of a Poisson distribution given that \( P(X=0) = 0.2 \).

### Step-by-Step Solution:

1. **Understand the Poisson Distribution**:
The probability mass function for a Poisson distribution is given by:
\[ P(X = n) = \frac{e^{-\lambda} \lambda^n}{n!} \]
where \( \lambda \) is the mean and variance of the distribution.

2. **Set Up the Equation for \( P(X=0) \)**:
We know that \( P(X=0) = 0.2 \). Using the formula for \( P(X=n) \) with \( n=0 \):
\[ P(X=0) = \frac{e^{-\lambda} \lambda^0}{0!} = e^{-\lambda} \]
Therefore, we can set up the equation:
\[ e^{-\lambda} = 0.2 \]

3. **Solve for \( \lambda \)**:
To find \( \lambda \), we take the natural logarithm of both sides:
\[ -\lambda = \ln(0.2) \]
Thus,
\[ \lambda = -\ln(0.2) \]

4. **Simplify \( \lambda \)**:
We can rewrite \( \ln(0.2) \):
\[ \ln(0.2) = \ln\left(\frac{1}{5}\right) = -\ln(5) \]
Therefore,
\[ \lambda = \ln(5) \]

5. **Determine the Variance**:
In a Poisson distribution, the variance is equal to the mean, which is \( \lambda \). Thus, the variance of the distribution is:
\[ \text{Variance} = \lambda = \ln(5) \]

### Final Answer:
The variance of the Poisson distribution is \( \ln(5) \).

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