In a Poisson distribution P(X=0) =2P(X=1) then the standard deviation =

Answer

To solve the problem step by step, we will use the properties of the Poisson distribution.

### Step 1: Understand the Poisson Distribution
The probability mass function of a Poisson distribution is given by:
\[ P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} \]
where \( \lambda \) is the average rate (mean) of occurrences in a fixed interval.

### Step 2: Set Up the Given Equation
We are given that:
\[ P(X = 0) = 2P(X = 1) \]
Using the formula for the Poisson distribution, we can write:
\[ P(X = 0) = \frac{e^{-\lambda} \lambda^0}{0!} = e^{-\lambda} \]
\[ P(X = 1) = \frac{e^{-\lambda} \lambda^1}{1!} = e^{-\lambda} \lambda \]

### Step 3: Substitute into the Equation
Substituting these into the given equation:
\[ e^{-\lambda} = 2 \cdot (e^{-\lambda} \lambda) \]

### Step 4: Simplify the Equation
We can simplify this equation by dividing both sides by \( e^{-\lambda} \) (assuming \( e^{-\lambda} \neq 0 \)):
\[ 1 = 2\lambda \]

### Step 5: Solve for \( \lambda \)
From the equation \( 1 = 2\lambda \), we can solve for \( \lambda \):
\[ \lambda = \frac{1}{2} \]

### Step 6: Find the Standard Deviation
The standard deviation \( \sigma \) of a Poisson distribution is given by:
\[ \sigma = \sqrt{\lambda} \]
Substituting the value of \( \lambda \):
\[ \sigma = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]

### Final Answer
The standard deviation is:
\[ \frac{1}{\sqrt{2}} \]

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