If X has Poisson distribution with `m=1`, then find `P(X le 1)` given `e^(-1) = 0.3678`.

Answer

To solve the problem of finding \( P(X \leq 1) \) for a Poisson distribution with parameter \( m = 1 \), we can follow these steps:

### Step 1: Understand the Poisson Probability Formula
The probability mass function for a Poisson distribution is given by:
\[ P(X = x) = \frac{m^x e^{-m}}{x!} \]
where \( m \) is the average rate (mean) of occurrence, \( e \) is the base of the natural logarithm, and \( x! \) is the factorial of \( x \).

### Step 2: Substitute the Given Values
In this case, \( m = 1 \). Therefore, the formula becomes:
\[ P(X = x) = \frac{1^x e^{-1}}{x!} \]
Since \( e^{-1} \) is given as \( 0.3678 \), we can write:
\[ P(X = x) = \frac{0.3678}{x!} \]

### Step 3: Calculate \( P(X = 0) \) and \( P(X = 1) \)
We need to find \( P(X \leq 1) \), which is the sum of \( P(X = 0) \) and \( P(X = 1) \).

1. **Calculate \( P(X = 0) \)**:
\[ P(X = 0) = \frac{0.3678}{0!} \]
Since \( 0! = 1 \):
\[ P(X = 0) = \frac{0.3678}{1} = 0.3678 \]

2. **Calculate \( P(X = 1) \)**:
\[ P(X = 1) = \frac{0.3678}{1!} \]
Since \( 1! = 1 \):
\[ P(X = 1) = \frac{0.3678}{1} = 0.3678 \]

### Step 4: Add the Probabilities
Now, we can find \( P(X \leq 1) \):
\[ P(X \leq 1) = P(X = 0) + P(X = 1) = 0.3678 + 0.3678 = 0.7356 \]

### Final Answer
Thus, the required probability \( P(X \leq 1) \) is:
\[ \boxed{0.7356} \]