To prove that $X$ follows a Poisson distribution with parameter $\mu$, we'll work through the definition and properties of the Poisson distribution.

Step 1: Define the Poisson distribution

A random variable $X$ follows a Poisson distribution with parameter $\mu$ (where $\mu > 0$) if its probability mass function (PMF) is given by: $P(X = k) = \frac{\mu^k e^{-\mu}}{k!}, \quad k = 0, 1, 2, \dots$

Step 2: Derive the mean and variance of the Poisson distribution

For the Poisson distribution, the mean $E(X)$ and variance $\text{Var}(X)$ are both equal to $\mu$. We will now show this.

Mean:

The mean of a Poisson-distributed random variable $X$ is given by: $E(X) = \sum_{k=0}^{\infty} k \cdot P(X = k)$ Substituting the PMF of the Poisson distribution: $E(X) = \sum_{k=0}^{\infty} k \cdot \frac{\mu^k e^{-\mu}}{k!}$

We can simplify this using the fact that $k \times \frac{1}{k!} = \frac{1}{(k-1)!}$: $E(X) = \mu e^{-\mu} \sum_{k=1}^{\infty} \frac{\mu^{k-1}}{(k-1)!}$ Changing the index of summation $j = k - 1$, we get: $E(X) = \mu e^{-\mu} \sum_{j=0}^{\infty} \frac{\mu^j}{j!}$ Recognize that the sum is the series expansion of $e^{\mu}$: $E(X) = \mu e^{-\mu} \cdot e^{\mu} = \mu$ So, the mean of the Poisson distribution is $\mu$.

Variance:

The variance of $X$ is given by: $\text{Var}(X) = E(X^2) - (E(X))^2$ First, calculate $E(X^2)$: $E(X^2) = \sum_{k=0}^{\infty} k^2 \cdot P(X = k)$ Expanding $k^2$ as $k(k-1) + k$, we get: $E(X^2) = \sum_{k=0}^{\infty} (k(k-1) + k) \cdot \frac{\mu^k e^{-\mu}}{k!}$ Splitting this into two sums, and simplifying, it can be shown that: $E(X^2) = \mu^2 + \mu$ Thus, the variance is: $\text{Var}(X) = E(X^2) - (E(X))^2 = \mu^2 + \mu - \mu^2 = \mu$ So, the variance of the Poisson distribution is also $\mu$.

Conclusion:

We have shown that for a Poisson-distributed random variable $X$ with parameter $\mu$, the mean and variance are both equal to $\mu$, which is a characteristic property of the Poisson distribution.

Would you like to dive into more details, or do you have any questions? Here are five related questions you might find helpful:

1. How is the Poisson distribution related to the binomial distribution?
2. Can you explain the relationship between the Poisson and exponential distributions?
3. How can you derive the Poisson distribution as a limit of the binomial distribution?
4. What are some real-world examples where the Poisson distribution is used?
5. How do you calculate the cumulative distribution function (CDF) for a Poisson-distributed variable?

Tip: The Poisson distribution is particularly useful for modeling the number of occurrences of an event over a fixed interval of time or space when the events are independent and occur at a constant average rate.