Poisson Distribution Quiz: Poisson Overview

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| Questions: 20 | Updated: Dec 17, 2025

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1) Which best defines a Poisson random variable X with rate λ over a fixed interval (time/space)?

The count of events occurring in the interval when events happen independently at a constant average rate λ

The time between two events in a process with rate λ

A measurement that can take any real value in an interval

The number of trials until first success with probability p

A Poisson random variable counts how many events occur in a fixed interval under the assumptions: independent occurrences, constant average rate λ, and events occur singly. This is why X takes values 0,1,2,… .

Explanation

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About This Quiz

Poisson Distribution Quiz: Poisson Overview - Quiz

Curious how the Poisson distribution models rare events? This quiz lets you explore how it predicts counts over time or space and why its shape changes with different rates. You’ll look at simple examples that reveal how Poisson patterns emerge everywhere. Dive in and see how well you can recognize... see morePoisson behavior.
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2) What is the probability mass function (pmf) of X ~ Poisson(λ)?

P(X = k) = e^{−λ} λ^k / k!, for k = 0,1,2,…

P(X = k) = λ^k, for k = 0,1,2,…

P(X = k) = (1 − λ)^k λ, for k = 0,1,2,…

P(X = k) = e^{−k} / λ!, for k = 0,1,2,…

By definition, the pmf of a Poisson variable with parameter λ > 0 is P(X = k) = e^{−λ} λ^k / k! for integer k ≥ 0; it sums to 1 because Σ_{k=0}^∞ e^{−λ} λ^k / k! = e^{−λ} e^{λ} = 1.

Explanation

By definition, the pmf of a Poisson variable with parameter λ > 0 is P(X = k) = e^{−λ} λ^k / k! for integer k ≥ 0; it sums to 1 because Σ_{k=0}^∞ e^{−λ} λ^k / k! = e^{−λ} e^{λ} = 1.

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3) For X ~ Poisson(λ), the mean and the variance are both equal to λ.

True

False

E[X] = λ and Var(X) = λ for a Poisson random variable. This follows from the mgf or by differentiating the probability generating function G_X(t) = exp(λ(t − 1)).

Explanation

E[X] = λ and Var(X) = λ for a Poisson random variable. This follows from the mgf or by differentiating the probability generating function G_X(t) = exp(λ(t − 1)).

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4) Select all core assumptions of a Poisson process that justify using a Poisson distribution for counts in a fixed interval.

Events occur independently of each other

The average rate λ is constant over the interval (stationary increments)

At most one event occurs at an instant (events occur singly)

Events arrive in large deterministic batches at fixed times

The probability of an event in a tiny interval is approximately proportional to the interval’s length

Poisson modeling assumes independence, a constant average rate λ, and that events occur singly. Also, for small Δt, P(1 event in Δt) ≈ λΔt and P(≥2 in Δt) is negligible. Batch arrivals (D) violate the 'singly' assumption.

Explanation

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5) Which scenario is MOST appropriate for a Poisson model?

The number of emails arriving to an inbox in one hour, assuming a roughly constant average rate

The time (in minutes) until the next bus arrives on a route with constant rate

Heights of students in a class

The number of coin flips until the first head

A Poisson model describes integer counts in a fixed interval with a constant average rate. The time until next event is modeled by an exponential distribution, heights are continuous and not counts, and 'until first head' is geometric.

Explanation

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6) If X ~ Poisson(2), what is P(X = 0)?

0.1353

0.2707

0.406

0.5

P(X=0) = e^{−λ} λ^0 / 0! = e^{−2} ≈ 0.135335283. Rounding to four decimals gives 0.1353.

Explanation

P(X=0) = e^{−λ} λ^0 / 0! = e^{−2} ≈ 0.135335283. Rounding to four decimals gives 0.1353.

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7) If X ~ Poisson(2), what is P(X = 3)?

0.0902

0.1353

0.1804

0.2707

P(X=3) = e^{−2} 2^3 / 3! = e^{−2} · 8 / 6 = e^{−2} · 1.333333… ≈ 0.135335283 × 1.333333… = 0.180447. Rounded → 0.1804.

Explanation

P(X=3) = e^{−2} 2^3 / 3! = e^{−2} · 8 / 6 = e^{−2} · 1.333333… ≈ 0.135335283 × 1.333333… = 0.180447. Rounded → 0.1804.

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8) Complete the Poisson pmf: P(X = k) = ________ for k = 0,1,2,… (parameter λ > 0).

The probability that X equals k events is e^{−λ} λ^k / k!, derived from the limit of the Binomial(n, p) with n → ∞, p → λ/n.

Explanation

The probability that X equals k events is e^{−λ} λ^k / k!, derived from the limit of the Binomial(n, p) with n → ∞, p → λ/n.

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9) A key property of Poisson processes is that events occur singly (the chance of two or more simultaneous events in an instant is negligible).

True

False

In the infinitesimal-interval characterization: P(1 event in Δt) ≈ λΔt and P(≥2 events in Δt) = o(Δt). Thus, simultaneous 'batch' events are negligible.

Explanation

In the infinitesimal-interval characterization: P(1 event in Δt) ≈ λΔt and P(≥2 events in Δt) = o(Δt). Thus, simultaneous 'batch' events are negligible.

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10) If X ~ Poisson(2) and Y ~ Poisson(5) are independent, what is the distribution of X + Y?

Poisson(7)

Poisson(10)

Binomial(7, 0.5)

Normal(μ=7, σ²=7) exactly

Sums of independent Poisson variables are Poisson with parameter equal to the sum of parameters: λ_total = 2 + 5 = 7. (Normal with μ=σ²=7 is an approximation for large λ, not exact.)

Explanation

Sums of independent Poisson variables are Poisson with parameter equal to the sum of parameters: λ_total = 2 + 5 = 7. (Normal with μ=σ²=7 is an approximation for large λ, not exact.)

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11) A call center receives on average λ = 4 calls per hour, modeled as Poisson. What is the expected number of calls in 30 minutes?

For half an hour, the mean scales with time: λ_30min = λ × (0.5) = 4 × 0.5 = 2. The Poisson parameter is proportional to the interval length.

Explanation

For half an hour, the mean scales with time: λ_30min = λ × (0.5) = 4 × 0.5 = 2. The Poisson parameter is proportional to the interval length.

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12) Let X ~ Poisson(1.5). What is P(X ≥ 1)?

11/13

0.6065

0.7769

0.9502

P(X ≥ 1) = 1 − P(0) = 1 − e^{−1.5} ≈ 1 − 0.223130 = 0.776870. Rounding to four decimals gives 0.7769.

Explanation

P(X ≥ 1) = 1 − P(0) = 1 − e^{−1.5} ≈ 1 − 0.223130 = 0.776870. Rounding to four decimals gives 0.7769.

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13) Which statements about Poisson distribution are true? Select all that apply.

E[X] = Var(X) = λ

Support is k = 0,1,2,…

The pmf sums to 1 over k = 0,1,2,…

It models time between events directly

It can approximate Binomial(n, p) when n is large and p is small with np = λ

The Poisson is a count distribution on nonnegative integers, with mean and variance λ and pmf summing to 1. It approximates Binomial when n large, p small, λ = np fixed. Time-between-events is exponential, not Poisson.

Explanation

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14) The Poisson distribution is typically used to model the time between events in a constant-rate process.

True

False

False. The time between events is modeled by the exponential distribution with rate λ. The Poisson models the count of events in a fixed interval.

Explanation

False. The time between events is modeled by the exponential distribution with rate λ. The Poisson models the count of events in a fixed interval.

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15) For λ = 3, P(X = 1) can be written compactly as ________.

Use the pmf with k=1: P(X=1) = e^{−3} 3^1 / 1! = 3e^{−3}.

Explanation

Use the pmf with k=1: P(X=1) = e^{−3} 3^1 / 1! = 3e^{−3}.

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16) If the average number of defects per meter is λ = 6 (Poisson), what is the variance of the number of defects per meter?

For Poisson, Var(X) = λ. With λ = 6, the variance is 6.

Explanation

For Poisson, Var(X) = λ. With λ = 6, the variance is 6.

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17) For X ~ Poisson(0.5), which count k is most probable (mode)?

K = 0

K = 1

K = 2

K = 3

For Poisson, the mode is ⌊λ⌋ (and also ⌊λ⌋ − 1 if λ is an integer). With λ = 0.5, ⌊0.5⌋ = 0, so k = 0 is the most probable.

Explanation

For Poisson, the mode is ⌊λ⌋ (and also ⌊λ⌋ − 1 if λ is an integer). With λ = 0.5, ⌊0.5⌋ = 0, so k = 0 is the most probable.

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18) Which situation violates a key assumption for Poisson modeling?

Accidents per hour increase sharply during rush hour (rate not constant)

Number of typos per printed page with stable editing process

Radioactive particle detections per minute with constant source strength

Customer arrivals per minute in a store during a steady period

Poisson assumes a constant average rate λ over the interval. A rush hour with sharply increasing rate violates stationarity.

Explanation

Poisson assumes a constant average rate λ over the interval. A rush hour with sharply increasing rate violates stationarity.

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19) For large λ, a Normal(μ = λ, σ² = λ) approximation can be used for Poisson probabilities.

True

False

By the central limit behavior for sums of rare independent events (or via De Moivre–Laplace), Poisson(λ) is well-approximated by Normal(λ, λ) when λ is large; continuity correction further improves accuracy.

Explanation

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20) If X ~ Poisson(λ1) and Y ~ Poisson(λ2) are independent, then X + Y is Poisson with parameter ________.

The sum of independent Poisson variables is Poisson with rate equal to the sum of the rates: λ_total = λ1 + λ2.

Explanation

The sum of independent Poisson variables is Poisson with rate equal to the sum of the rates: λ_total = λ1 + λ2.

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