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The given problem asks for probabilities in a Poisson distribution for different values of \(\lambda\) and specific ranges of \(X\). The Poisson probability mass function (PMF) is given by:

\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
\]

To find the cumulative probabilities for the specified cases, we need to sum the individual Poisson probabilities.

Let's break this down case by case:

### a) Let \(\lambda = 4.0\), find \(P(X \geq 3)\)

The probability \(P(X \geq 3)\) can be written as:

\[
P(X \geq 3) = 1 - P(X \leq 2)
\]

So we first calculate \(P(X \leq 2)\), which is the sum of \(P(X = 0)\), \(P(X = 1)\), and \(P(X = 2)\).

### b) Let \(\lambda = 0.4\), find \(P(X \leq 1)\)

This requires calculating:

\[
P(X \leq 1) = P(X = 0) + P(X = 1)
\]

### c) Let \(\lambda = 7.0\), find \(P(X \leq 2)\)

Here, we calculate:

\[
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
\]

### d) Let \(\lambda = 3.7\), find \(P(X \geq 2)\)

This can be written as:

\[
P(X \geq 2) = 1 - P(X \leq 1)
\]

### e) Let \(\lambda = 5.4\), find \(P(X \leq 2)\)

Similar to (c), we calculate:

\[
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
\]

I'll compute these values now.Here are the results for the given Poisson distribution problems:

- **a)** \(\lambda = 4.0\), \(P(X \geq 3) = 0.7619\)
- **b)** \(\lambda = 0.4\), \(P(X \leq 1) = 0.9384\)
- **c)** \(\lambda = 7.0\), \(P(X \leq 2) = 0.0296\)
- **d)** \(\lambda = 3.7\), \(P(X \geq 2) = 0.8838\)
- **e)** \(\lambda = 5.4\), \(P(X \leq 2) = 0.0948\)

Would you like a detailed explanation of any of these solutions or further questions about Poisson distributions?

### Related Questions:
1. What is the general formula for the Poisson distribution, and why is \(\lambda\) important?
2. How would \(P(X = k)\) change if \(\lambda\) were doubled for any of the examples?
3. Can you compute \(P(X = k)\) for non-integer values of \(k\)?
4. What is the relationship between the Poisson and normal distributions when \(\lambda\) becomes large?
5. How does the cumulative distribution function (CDF) work in the context of the Poisson distribution?

#### Tip:
In problems involving \(P(X \geq k)\), it's often easier to calculate \(1 - P(X \leq k-1)\) rather than summing all probabilities for \(P(X = k)\) and above.

Assume a Poisson distribution. Find the probabilities for various values of lambda and specific ranges of X.

Explore how to calculate probabilities using the Poisson distribution for different values of lambda and X. This guide provides step-by-step solutions and formulas used in calculating cumulative probabilities, suitable for university-level students.

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