We are given a probability mass function (PMF) for a discrete random variable $X$ that generates ordinal data:

$f(k) = \frac{\lambda^k e^{-\lambda}}{k!} \quad \text{where} \quad \lambda = 1, \, k \in \mathbb{N}$

This is the PMF of a Poisson distribution with parameter $\lambda = 1$.

We are tasked with finding $P(X > 2)$, which represents the probability that the random variable $X$ takes values greater than 2. The cumulative probability $P(X > 2)$ can be calculated as:

$P(X > 2) = 1 - P(X \leq 2)$

Where $P(X \leq 2)$ is the cumulative probability for $X = 0, 1, 2$. Let's compute this step by step:

Step 1: Compute $P(X \leq 2)$

The cumulative probability $P(X \leq 2)$ is the sum of probabilities for $X = 0, 1, 2$:

$P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)$

Using the Poisson distribution PMF:

• $P(X = 0) = \frac{1^0 e^{-1}}{0!} = e^{-1}$
• $P(X = 1) = \frac{1^1 e^{-1}}{1!} = e^{-1}$
• $P(X = 2) = \frac{1^2 e^{-1}}{2!} = \frac{e^{-1}}{2}$

Now, sum these probabilities:

$P(X \leq 2) = e^{-1} + e^{-1} + \frac{e^{-1}}{2} = e^{-1} \left(1 + 1 + \frac{1}{2}\right) = e^{-1} \times \frac{5}{2}$

Thus:

$P(X \leq 2) = \frac{5}{2} e^{-1}$

Step 2: Compute $P(X > 2)$

Now, use the complement rule:

$P(X > 2) = 1 - P(X \leq 2) = 1 - \frac{5}{2} e^{-1}$

Final Answer:

$P(X > 2) = 1 - \frac{5}{2} e^{-1}$

This is the exact expression for $P(X > 2)$. If you want a numerical approximation, you can plug in the value of $e^{-1} \approx 0.3679$:

$P(X > 2) \approx 1 - \frac{5}{2} \times 0.3679 = 1 - 0.91975 \approx 0.08025$

Summary:

$P(X > 2) \approx 0.08025$

Would you like further clarification or a deeper dive into any part of this calculation?

Additional Questions:

1. How does the Poisson distribution relate to other common distributions like the binomial or normal distribution?
2. How would you compute $P(X = k)$ for other values of $k$?
3. What is the expected value and variance of a Poisson distribution with $\lambda = 1$?
4. How would you use this distribution to model events occurring at a fixed rate over time?
5. What is the cumulative distribution function (CDF) of a Poisson distribution?

Tip:

For quick calculations, remember that the Poisson distribution is often used to model rare events happening at a constant average rate, especially in contexts like traffic flows, call centers, and biological processes.