In a Poisson distribution if `P(X=0)=P(X=1)=k`, the value of k is

Options

A. 1
B. `1//e`
C. e
D. `sqrte`

Answer

To solve the problem, we need to find the value of \( k \) given that in a Poisson distribution, \( P(X=0) = P(X=1) = k \).

### Step-by-Step Solution:

1. **Understand the Poisson Distribution Formula**:
The probability mass function for a Poisson distribution is given by:
\[ P(X = x) = \frac{e^{-\mu} \mu^x}{x!} \]
where \( \mu \) is the average rate (mean) of occurrences.

2. **Set Up the Equations**:
We know that:
\[ P(X = 0) = P(X = 1) = k \]
Using the Poisson formula, we can express these probabilities:
\[ P(X = 0) = \frac{e^{-\mu} \mu^0}{0!} = e^{-\mu} \]
\[ P(X = 1) = \frac{e^{-\mu} \mu^1}{1!} = e^{-\mu} \mu \]

3. **Equate the Two Probabilities**:
Since \( P(X = 0) = P(X = 1) \), we can set the two equations equal to each other:
\[ e^{-\mu} = e^{-\mu} \mu \]

4. **Simplify the Equation**:
We can divide both sides by \( e^{-\mu} \) (noting that \( e^{-\mu} \neq 0 \)):
\[ 1 = \mu \]

5. **Find the Value of \( k \)**:
Now that we have \( \mu = 1 \), we can substitute this back into the equation for \( P(X = 0) \):
\[ P(X = 0) = e^{-1} = \frac{1}{e} \]
Thus, \( k = \frac{1}{e} \).

### Final Answer:
The value of \( k \) is \( \frac{1}{e} \).

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