When n is very large and p is very small in the binomial distribution, then X follows the Poisson distribution with parameter m =

Answer

To solve the question, we need to establish the relationship between the binomial distribution and the Poisson distribution under the given conditions.

### Step-by-Step Solution:

1. **Understand the Binomial Distribution**:
The binomial distribution is defined as \( X \sim B(n, p) \), where \( n \) is the number of trials and \( p \) is the probability of success in each trial.

2. **Conditions Given**:
We are given that \( n \) is very large (approaching infinity) and \( p \) is very small (approaching zero). This means we are looking at a scenario where the number of trials is significantly high, but the probability of success in each trial is very low.

3. **Product of n and p**:
We need to consider the product \( n \cdot p \). As \( n \) increases and \( p \) decreases, we want to find a constant value for \( m \).

4. **Define Parameter m**:
We define \( m \) as the product of \( n \) and \( p \):
\[ m = n \cdot p \]
This product remains constant even as \( n \) becomes very large and \( p \) becomes very small.

5. **Convergence to Poisson Distribution**:
Under these conditions (large \( n \) and small \( p \)), the binomial distribution can be approximated by the Poisson distribution. Specifically, if \( n \) is large and \( p \) is small such that \( m = n \cdot p \) is constant, then:
\[ X \sim \text{Poisson}(m) \]

6. **Final Answer**:
Therefore, the parameter \( m \) for the Poisson distribution is given by:
\[ m = n \cdot p \]

### Summary:
When \( n \) is very large and \( p \) is very small, the binomial distribution \( X \sim B(n, p) \) can be approximated by the Poisson distribution with parameter \( m = n \cdot p \).

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