For the Poisson distribution

Options

A. Mean= E(X) = m
B. Var(X) = m
C. Mean = E(X) = m and Var(X) = m
D. Mean = `E(X) != m and Var (X) = m.`

Answer

To solve the question regarding the Poisson distribution, we need to analyze the properties of the distribution, particularly focusing on the relationship between the mean and variance.

### Step-by-Step Solution:

1. **Understanding the Poisson Distribution**:
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event.

2. **Mean and Variance of Poisson Distribution**:
For a Poisson distribution with parameter \( \lambda \):
- The mean \( E(X) \) is given by \( \lambda \).
- The variance \( Var(X) \) is also given by \( \lambda \).

3. **Setting Up the Relationships**:
From the properties of the Poisson distribution, we can establish the following relationships:
- Mean \( E(X) = \lambda \)
- Variance \( Var(X) = \lambda \)

4. **Conclusion**:
Since both the mean and variance are equal and both are represented by the parameter \( \lambda \), we can conclude that:
\[ E(X) = Var(X) = \lambda \]
This means that the mean is equal to the variance in a Poisson distribution.

5. **Identifying the Correct Option**:
Given the options:
- Option 1: Mean = \( E(X) = M \)
- Option 2: \( X = M \)
- Option 3: Mean = \( E(X) = M \) and Variance of \( X = M \)
- Option 4: Mean = \( E(X) \neq M \) and Variance of \( X = M \)

The correct option that matches our conclusion is **Option 3**, which states that the mean is equal to \( E(X) \) and the variance of \( X \) is also equal to \( M \).

### Final Answer:
**Option 3: Mean = \( E(X) = M \) and Variance of \( X = M \)**