343 questions · 33 question types identified
Questions that only ask to state, list, or write down the conditions/assumptions needed for a Poisson distribution to be valid, without requiring explanation or contextual application.
Questions that require explaining what specific conditions mean in the given context, commenting on validity of assumptions, or discussing whether conditions are likely to hold in a particular scenario.
Questions asking for P(X = k), P(X ≤ k), P(X ≥ k), or P(a ≤ X < b) for a Poisson variable over a single given time period with no scaling required.
Questions involving the sum of exactly two independent Poisson random variables, typically asking for probabilities of the combined total.
A question is this type if and only if it requires conducting a formal hypothesis test about the Poisson parameter λ, typically testing whether the rate has changed.
Questions that use normal approximation to find probability for a Poisson distribution over a single time period (e.g., one year, one day, one hour), without comparing multiple periods or using binomial structure.
A question is this type if and only if it requires finding the parameter λ given a probability condition such as P(X = k) equals a specific value or P(X = a) = P(X = b).
Questions that provide sample mean and variance (or standard deviation) and ask whether these support a Poisson model by comparing their closeness, since for Poisson distributions mean equals variance.
Questions that first use normal approximation for a Poisson distribution, then apply this to multiple independent periods using binomial probability (e.g., probability that x out of n weeks satisfy a condition).
A question is this type if and only if it asks to find E(Y) and/or Var(Y) where Y is a linear function of Poisson random variables or involves sums/differences of Poisson variables.
A question is this type if and only if it involves using a chi-squared goodness-of-fit test to determine whether observed frequency data fits a Poisson distribution.
A question is this type if and only if it involves both Poisson and binomial distributions in the same problem, typically where Poisson events are followed by binomial trials.
Questions that involve calculating probabilities across multiple independent time periods (e.g., probability that events occur in each of several separate periods, or only one of several periods).
Questions where independent Poisson variables are first scaled to different time periods before summing, requiring rate adjustment before applying the sum property.
Questions requiring scaling of the Poisson mean to a different time period (e.g., from per week to per 2 weeks, or per minute to per 90 minutes) before calculating probabilities.
Questions asking for the probability that specific events occur in two or more independent Poisson processes separately (e.g., at least 2 men AND at least 3 women), requiring multiplication of individual Poisson probabilities.
Questions asking about the total count across multiple independent Poisson processes, using the result that the sum of independent Poisson variables is also Poisson with mean equal to the sum of individual means.
A question is this type if and only if it asks for P(A|B) where both events involve Poisson random variables, requiring use of conditional probability formula.
A question is this type if and only if it requires algebraic proof of Poisson distribution properties such as E(X), Var(X), or E(X(X-1)) using the probability mass function.
Questions involving the sum of three or more independent Poisson random variables, requiring combination of multiple sources or types.
Questions requiring the smallest n such that the probability of zero occurrences falls below a given threshold (e.g., P(X=0) < 0.05 or P(X=0) < 0.01).
Questions that provide a frequency table of observed counts and ask whether the data support a Poisson model, typically requiring calculation of sample mean and variance from the frequency distribution.
| No of errors | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| No of pages | 16 | 38 | 41 | 29 | 17 | 7 | 2 |
Questions that ask to use Poisson approximation to approximate a binomial distribution B(n,p) where n is large and p is small, with straightforward probability calculations.
Questions specifically about consecutive or successive time periods where the temporal ordering or sequence matters (e.g., first half vs second half, successive minutes).
Questions requiring the minimum number of items to stock so that the probability of meeting demand exceeds a threshold (practical inventory problems).
Questions requiring the largest n such that the probability of zero occurrences exceeds a given threshold (e.g., P(X=0) > 0.9 or P(X=0) ≥ 0.95).
Questions requiring the smallest n such that the probability of at least k occurrences satisfies a condition (e.g., P(X≥1) = 0.9, or P(X≥n) < 0.05).
Questions that explicitly ask to justify why Poisson approximation is suitable, explain why both binomial and Poisson could be used, or compare distributions.
A question is this type if and only if it involves finding probabilities related to waiting times or the number of trials until a Poisson event occurs.
A question is this type if and only if it asks about the distribution of the sample mean from multiple independent Poisson observations, typically using CLT for large samples.
Questions that require scaling the Poisson parameter to a single different time period and calculating probabilities for that period only.
Questions involving conditional probabilities or competitive outcomes between processes (e.g., one team scoring more than another, overflow between guesthouses), requiring consideration of relative outcomes rather than just joint or total counts.
Questions that require finding an unknown parameter (typically n) given a probability condition, using the Poisson approximation to binomial.