Poisson Tables Quiz: Constructing And Reading Poisson Tables

1) What does a Poisson probability table typically show?

Probabilities P(X = k) for various k values given λ

Cumulative probabilities only

Random samples of Poisson events

Normal distribution probabilities

A Poisson probability table lists the probability P(X = k) = e^{−λ} λ^k / k! for multiple k values at fixed λ values, allowing easy lookup instead of manual computation.

Explanation

A Poisson probability table lists the probability P(X = k) = e^{−λ} λ^k / k! for multiple k values at fixed λ values, allowing easy lookup instead of manual computation.

2) In a Poisson probability table for λ = 3, what does the row labeled k = 2 represent?

The probability of exactly two events occurring in the given interval

The mean of the distribution

The cumulative probability of up to two events

The variance of the distribution

Each row in a Poisson table represents a specific count k. For k = 2, it gives P(X = 2) = e^{−3} 3^2 / 2! = 0.2240, meaning exactly two events occur when the mean rate is 3.

Explanation

Each row in a Poisson table represents a specific count k. For k = 2, it gives P(X = 2) = e^{−3} 3^2 / 2! = 0.2240, meaning exactly two events occur when the mean rate is 3.

3) If a Poisson table shows P(X = 3) = 0.180 for λ = 2, what does this value mean?

There is an 18% chance of exactly three events occurring when the mean rate is 2

There is an 18% chance of three or more events

There is an 18% chance of at least three events

There is an 18% chance of fewer than three events

For λ = 2, P(X = 3) = e^{−2} 2^3 / 3! = 0.1804, meaning exactly three events occur with that probability, consistent with the value shown in the table.

Explanation

For λ = 2, P(X = 3) = e^{−2} 2^3 / 3! = 0.1804, meaning exactly three events occur with that probability, consistent with the value shown in the table.

4) How are Poisson tables constructed for multiple λ values?

Each column corresponds to a different λ, with rows listing P(X = k) values for each k

Each row corresponds to a λ value and each column to a k value

The tables only show one λ at a time

They include negative λ for comparison

Poisson tables are structured with rows as event counts k = 0,1,2,… and columns as different λ values, showing probabilities P(X = k; λ).

Explanation

Poisson tables are structured with rows as event counts k = 0,1,2,… and columns as different λ values, showing probabilities P(X = k; λ).

5) Poisson tables can be used to approximate Binomial probabilities when n is large and p is small (with λ = np).

True

False

When n is large and p is small, the Binomial(n,p) distribution converges to Poisson(λ = np). Tables make approximations quicker without full binomial expansion.

Explanation

When n is large and p is small, the Binomial(n,p) distribution converges to Poisson(λ = np). Tables make approximations quicker without full binomial expansion.

6) If a Poisson table for λ = 5 shows P(X ≤ 2) = 0.1247, what does that represent?

The cumulative probability of observing 2 or fewer events

The probability of exactly two events

The expected number of events

The probability of more than two events

Cumulative entries list the total probability up to k = 2: P(X ≤ 2) = Σ_{k=0}^{2} P(X = k). The complement 1 − 0.1247 = 0.8753 gives P(X ≥ 3).

Explanation

Cumulative entries list the total probability up to k = 2: P(X ≤ 2) = Σ_{k=0}^{2} P(X = k). The complement 1 − 0.1247 = 0.8753 gives P(X ≥ 3).

7) If λ = 6, what is the most probable count (mode) according to the Poisson table?

5

6

7

8

The Poisson mode is ⌊λ⌋ or (λ − 1) if λ is integer. For λ = 6, both 5 and 6 are modes since λ is an integer, but typically tables highlight 6 as the peak probability.

Explanation

The Poisson mode is ⌊λ⌋ or (λ − 1) if λ is integer. For λ = 6, both 5 and 6 are modes since λ is an integer, but typically tables highlight 6 as the peak probability.

8) What is the relationship between λ and the Poisson table shape?

As λ increases, the table’s probabilities shift right and spread wider

As λ increases, all probabilities shrink toward zero

As λ increases, the distribution becomes left-skewed

As λ increases, only P(X = 0) increases

Larger λ increases the mean and variance (both equal λ), shifting the distribution’s mass toward higher k values and spreading it out more, visible in Poisson tables.

Explanation

Larger λ increases the mean and variance (both equal λ), shifting the distribution’s mass toward higher k values and spreading it out more, visible in Poisson tables.

9) The probabilities in a Poisson probability table always sum to 1 across all k values for a given λ.

True

False

Since P(X = k) = e^{−λ} λ^k / k! for k = 0 to ∞, the total sum equals e^{−λ} Σ λ^k / k! = e^{−λ} e^{λ} = 1, confirming normalization.

Explanation

Since P(X = k) = e^{−λ} λ^k / k! for k = 0 to ∞, the total sum equals e^{−λ} Σ λ^k / k! = e^{−λ} e^{λ} = 1, confirming normalization.

10) If λ = 2 and P(X = 0) = 0.1353, what is P(X ≥ 1)?

11/13

0.2707

0.5

0.1353

P(X ≥ 1) = 1 − P(0) = 1 − 0.1353 = 0.8647. In a Poisson table, cumulative complements are easily derived from the first few rows.

Explanation

P(X ≥ 1) = 1 − P(0) = 1 − 0.1353 = 0.8647. In a Poisson table, cumulative complements are easily derived from the first few rows.

11) As λ increases, the Poisson distribution becomes more symmetric.

True

False

At small λ, the Poisson distribution is right-skewed. As λ grows, the shape approaches symmetry and resembles a normal distribution centered at λ.

Explanation

At small λ, the Poisson distribution is right-skewed. As λ grows, the shape approaches symmetry and resembles a normal distribution centered at λ.

12) Which steps describe constructing a Poisson table from scratch for a given λ?

Choose k values (e.g., 0 to 10)

Compute P(X = k) = e^{−λ} λ^k / k! for each k

Round probabilities to a consistent number of decimals

List them sequentially to verify ΣP(X = k) ≈ 1

Skip large k values even if probabilities remain high

To construct the table, calculate each P(X = k) sequentially for k = 0,… until probabilities are negligible. Rounding consistency ensures readability, and the sum check ensures validity.

Explanation

To construct the table, calculate each P(X = k) sequentially for k = 0,… until probabilities are negligible. Rounding consistency ensures readability, and the sum check ensures validity.

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13) If a Poisson table lists P(X ≤ 4) = 0.7851 for λ = 5, what is P(X > 4)?

0.2149

0.3

0.5

0.7851

Since total probability equals 1, P(X > 4) = 1 − P(X ≤ 4) = 1 − 0.7851 = 0.2149. Poisson tables simplify such complement calculations.

Explanation

Since total probability equals 1, P(X > 4) = 1 − P(X ≤ 4) = 1 − 0.7851 = 0.2149. Poisson tables simplify such complement calculations.

14) Select all key components of a Poisson probability table.

λ (mean number of events per interval)

K values (event counts)

P(X = k) probabilities

Cumulative probabilities P(X ≤ k)

Standard deviation of λ

A Poisson table usually includes λ, discrete k values, individual probabilities P(X = k), and sometimes cumulative probabilities for convenience. Standard deviation is not typically listed.

Explanation

A Poisson table usually includes λ, discrete k values, individual probabilities P(X = k), and sometimes cumulative probabilities for convenience. Standard deviation is not typically listed.

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15) Poisson tables provide probabilities for both positive and negative k values.

True

False

Poisson-distributed counts are nonnegative integers (k = 0,1,2,…). Negative counts are impossible and excluded from all tables.

Explanation

Poisson-distributed counts are nonnegative integers (k = 0,1,2,…). Negative counts are impossible and excluded from all tables.

16) When reading a Poisson probability table, which steps help find P(X ≥ 3)?

Locate λ for the problem

Find P(X = 0), P(X = 1), and P(X = 2)

Add these probabilities and subtract from 1

Look directly at P(X = 3)

Ignore λ since it doesn’t affect results

To find P(X ≥ 3), use the complement: 1 − [P(X ≤ 2)]. The table gives P(X = 0), P(X = 1), and P(X = 2), which you sum and subtract from 1.

Explanation

To find P(X ≥ 3), use the complement: 1 − [P(X ≤ 2)]. The table gives P(X = 0), P(X = 1), and P(X = 2), which you sum and subtract from 1.

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17) If λ = 4, what is the cumulative probability P(X ≤ 3)?

0.4335

0.5665

0.7619

0.8153

From Poisson tables, for λ = 4: P(X ≤ 3) = 0.5665. This is the sum of P(0)=0.0183, P(1)=0.0733, P(2)=0.1465, and P(3)=0.2340.

Explanation

From Poisson tables, for λ = 4: P(X ≤ 3) = 0.5665. This is the sum of P(0)=0.0183, P(1)=0.0733, P(2)=0.1465, and P(3)=0.2340.

18) For λ = 4, the probability P(X = 0) from the Poisson formula is ________.

Using the formula P(X = 0) = e^{−λ} λ^0 / 0! = e^{−4}. This represents the probability of zero events when the average is 4 per interval.

Explanation

Using the formula P(X = 0) = e^{−λ} λ^0 / 0! = e^{−4}. This represents the probability of zero events when the average is 4 per interval.

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19) For λ = 3, P(X = 1) = ________ (in exact symbolic form).

P(X=1) = e^{−3} 3^1 / 1! = 3e^{−3}, representing the probability of one event with mean rate 3.

Explanation

P(X=1) = e^{−3} 3^1 / 1! = 3e^{−3}, representing the probability of one event with mean rate 3.

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20) When λ = 2, the cumulative probability up to k = 3 is ________ (rounded to four decimals).

Using table values: P(0)=0.1353, P(1)=0.2707, P(2)=0.2707, P(3)=0.1804; sum = 0.8571. So P(X ≤ 3)=0.8571.

Explanation

Using table values: P(0)=0.1353, P(1)=0.2707, P(2)=0.2707, P(3)=0.1804; sum = 0.8571. So P(X ≤ 3)=0.8571.

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Poisson Tables Quiz: Constructing and Reading Poisson Tables