The number of traffic tickets issued follows a Poisson distribution, where the average number of tickets per month ($\lambda$) is 5.6. The probability mass function (PMF) for the Poisson distribution is given by:

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

where:

• $\lambda$ is the average number of events (5.6 in this case),
• $k$ is the number of occurrences for which we want to find the probability, and
• $e$ is Euler's number ($e \approx 2.71828$).

Part (a)

We want to find the probability that exactly 5 traffic tickets will be issued. This is $P(X = 5)$.

$P(X = 5) = \frac{5.6^5 e^{-5.6}}{5!}$

Part (b)

We want to find the probability that 3 or fewer traffic tickets will be issued. This is $P(X \leq 3)$, which we compute as:

$P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

Each of these probabilities will be calculated using the Poisson formula:

$P(X = k) = \frac{5.6^k e^{-5.6}}{k!}$

for $k = 0, 1, 2, 3$.

Part (c)

We want to find the probability that more than 6 traffic tickets will be issued. This is $P(X > 6)$, which can be calculated as:

$P(X > 6) = 1 - P(X \leq 6)$

Where $P(X \leq 6)$ can be calculated as:

$P(X \leq 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)$

Let me calculate each part for you.Here are the results for the given Poisson distribution problem:

Part (a)

The probability that exactly 5 traffic tickets will be issued next month is approximately:

$P(X = 5) \approx 0.1697$

Part (b)

The probability that 3 or fewer traffic tickets will be issued next month is approximately:

$P(X \leq 3) \approx 0.1906$

Part (c)

The probability that more than 6 traffic tickets will be issued next month is approximately:

$P(X > 6) \approx 0.3297$

Let me know if you need more details or further clarifications!

Follow-up questions:

1. How does the shape of the Poisson distribution change with varying $\lambda$?
2. Can you calculate the variance and standard deviation for this Poisson distribution?
3. What is the expected number of tickets issued in 3 months if the average remains the same?
4. How does the Poisson distribution compare to the normal distribution for large $\lambda$?
5. How would the probabilities change if the average number of tickets was 7 instead of 5.6?

Tip:

For small $\lambda$, the Poisson distribution is skewed to the right, but as $\lambda$ increases, it begins to resemble a normal distribution.