Hey there! Let's break down these three Poisson distribution functions in R—they all work with the same distribution but serve completely distinct purposes. I'll walk through each one with clear definitions and real-world use cases so you know exactly when to reach for which.

dpois(): The Probability Mass Function (PMF) #

Think of dpois() as your "exact count probability calculator." It returns the probability that a Poisson-distributed random variable equals a specific integer value x.

The syntax is straightforward:

dpois(x, lambda)

• x: The number of events you're interested in (must be a non-negative integer)
• lambda: The average rate of events per unit time/space (the core parameter of the Poisson distribution)

When to use it: #

• You need the probability of exactly k events occurring. For example: "If my coffee shop gets an average of 5 customers per 10 minutes, what's the chance exactly 7 customers walk in during that window?" Use dpois(7, lambda=5) to get your answer.
• You're plotting the shape of a Poisson distribution. Generate a sequence of x values, pair them with dpois() outputs, and you've got a probability bar chart.

ppois(): The Cumulative Distribution Function (CDF) #

ppois() is all about cumulative probabilities. It gives you the chance that the number of events is less than or equal to a given value q. To get the probability of more than q events, just subtract the result from 1: 1 - ppois(q, lambda).

Syntax:

ppois(q, lambda)

• q: The maximum number of events you're considering (again, non-negative integer)
• lambda: Same average rate parameter as above

When to use it: #

• You want the probability of at most k events. Example: "What's the likelihood my coffee shop gets 3 or fewer customers in 10 minutes (with lambda=5)?" Run ppois(3, lambda=5) to find out.
• Hypothesis testing or statistical inference. If you observe an unusually high number of events (say 10 customers in 10 minutes), you can calculate the p-value as 1 - ppois(9, lambda=5) to see how rare that observation is.

rpois(): The Random Number Generator #

As the name suggests, rpois() spits out random numbers that follow a Poisson distribution. It's your go-to for simulating real-world scenarios that fit the Poisson model.

Syntax:

rpois(n, lambda)

• n: The number of random values you want to generate
• lambda: The average event rate

When to use it: #

• Simulating data: Suppose you need to create a dataset of 1000 10-minute intervals showing customer counts at your shop (with lambda=5). rpois(1000, lambda=5) will generate that dataset for you.
• Monte Carlo simulations: If you're testing how a statistical model performs under Poisson-distributed noise, or estimating a complex probability that's hard to calculate analytically, rpois() lets you generate thousands of samples to run your simulation.

Quick Example to Tie It All Together #

Let's use lambda=5 (average 5 customers per 10 minutes) for all three:

• dpois(5, 5) → Returns ~0.1755 (probability of exactly 5 customers)
• ppois(5, 5) → Returns ~0.6160 (probability of 5 or fewer customers)
• rpois(4, 5) → Might output something like [3, 6, 4, 7] (4 random customer counts, each following the Poisson(5) distribution—results vary each run!)

内容的提问来源于stack exchange，提问作者manu