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Poisson Distribution

The Poisson distribution counts how many random events happen in a fixed time or space when they come at a steady average rate λ and independently of each other. P(X = k) = e^(−λ) λ^k / k!. Its mean and its variance are both λ. It is also a good shortcut for a binomial with large n and small p, with λ = np.

🎬 Step-by-step story

  1. A quiet road. We will count the cars that pass in each minute. No car has come yet.
  2. Cars arrive at random times. Each red dot is one car. Nobody knows when the next one will come.
  3. Look at one 1-minute window (yellow). We count the cars inside it. This count is called k.
  4. Now the blue bars. Each bar shows how likely a count is: k = 0, 1, 2 and so on. Tall bars are the likely counts.
  5. We make the traffic heavier. The average λ goes from 2 to 6. The bars slide to the right and spread out.
  6. Free play. Move λ and k. See that the average and the spread are both λ. Press New hour to get new random cars.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why can k be any number, even very large?

There is no fixed number of trials. In theory any count can happen, but big counts have very tiny bars.

Does a Poisson process remember the last car?

No. Events are independent, so a car arriving now does not change when the next one comes.

Why does the 1-minute window matter?

λ is for that window size. Make the window longer and λ grows in the same proportion.

What happens to the shape when λ grows?

It moves right and spreads, and looks more like a bell curve.

Why does each New hour look different?

The events are random. The bars show chances, not what happens in one hour.

What is a Poisson distribution?

Sometimes we do not count successes in a fixed number of trials. We count how many events happen in a fixed time or space: cars in a minute, calls in an hour, typos on a page.

If events happen at random, one at a time, independently, and at a steady average rate, the count X follows a Poisson distribution with parameter λ (lambda). λ is the average number of events in the slot you chose.

The formula

P(X = k) = e−λ · λk / k! for k = 0, 1, 2, 3, …

Here e ≈ 2.718 and k! means k × (k−1) × … × 1 (with 0! = 1). Example: λ = 2, then P(0) = e−2 = 0.135 and P(2) = e−2 × 4/2 = 0.271. All the probabilities add up to 1.

Change the slot, change λ. If 3 calls come per hour, then in 2 hours λ = 6.

Mean, variance and the shape

The mean is E(X) = λ and the variance is Var(X) = λ, so the standard deviation is √λ. This is a quick test: if the average count and the variance of real data are close, Poisson is a good model.

For small λ the bars lean to the left (a long tail on the right). When λ grows, the bars move right, spread out, and look more and more like a bell curve (the normal distribution).

Poisson as a shortcut for binomial

When n is large and p is small (for example n = 200, p = 0.02), the binomial B(n, p) is hard to compute. Use Poisson with λ = np (here 4). A thumb rule is n ≥ 50 and np ≤ 5 or so.

Try it

Count something at home for 10 slots of 1 minute: cars from your window, or messages on your phone. Find the average. Now guess: will a slot with 0 events be common? Check with P(0) = e−λ. In the 3D, press New hour and see that each hour looks different, but the bars stay the same.

Key formulas and definitions

Worked examples

1. Cars pass a gate at an average of 2 per minute. Find the probability that no car passes in a minute.

λ = 2. P(0) = e^(−2) × 2^0 / 0! = e^(−2) = 0.135.

2. A help desk gets 3 calls per hour on average. Find P(exactly 2 calls in an hour).

λ = 3. P(2) = e^(−3) × 3² / 2! = 0.0498 × 9 / 2 = 0.224.

3. A shop sees 4 customers per 10 minutes. Find P(at most 1 customer in 10 minutes).

λ = 4. P(0) + P(1) = e^(−4)(1 + 4) = 0.0183 × 5 = 0.092.

4. A page has on average 1 typing error. Find P(at least one error).

λ = 1. P(X ≥ 1) = 1 − P(0) = 1 − e^(−1) = 1 − 0.368 = 0.632.

5. Calls come at 3 per hour. Find P(exactly 6 calls in 2 hours).

In 2 hours λ = 6. P(6) = e^(−6) × 6⁶ / 6! = 0.002479 × 46656 / 720 = 0.161.

6. A bulb factory has a 2% defect rate. In a box of 200 bulbs find P(exactly 3 defective) using Poisson.

n = 200, p = 0.02, so λ = np = 4. P(3) = e^(−4) × 4³ / 3! = 0.0183 × 64 / 6 = 0.195.

Common mistakes

Practice quiz

1. The mean of a Poisson distribution with parameter λ is:
2. The variance of a Poisson distribution is:
3. For λ = 2, P(X = 0) is:
4. Which is a good Poisson situation?
5. 3 events per hour on average. In 2 hours λ is:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is the Poisson distribution in simple words?

It tells the chance of seeing exactly k random events in a fixed time or space when events come at a steady average rate λ.

Why are mean and variance equal in Poisson?

Because rare random events add spread exactly as fast as they add average. This is a special property of the law: both equal λ.

When do I use Poisson instead of binomial?

Use Poisson when you count events in a slot with no fixed number of trials, or as a shortcut for binomial when n is large and p is small.

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