23-Ind-A6 Systems Simulation · Undated paper

Question 5 of 10: Distribution Concepts — Fill in the Blanks

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Question 5 (Part A.5): Distribution Concepts — Fill in the Blanks (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

This is a definitional/concepts question with no numeric calculation — each blank is answered directly from the standard input-modelling taxonomy used throughout simulation practice.

1. (a) Normal distribution. Models continuous quantities that are the additive result of many small, independent effects around a central value — e.g. measurement/estimation errors, and physical quantities like dimensions, weights, or times that cluster symmetrically around a mean (the Central Limit Theorem is the underlying justification).
2. (b) Sum of exponentials. The sum of $k$ i.i.d. exponential random variables is $\boxed{\text{Erlang}(k,\lambda)}$, a special case of the Gamma distribution with integer shape parameter.
3. (c) Lognormal fitting. Parameters are fit by $\boxed{\text{taking the natural logarithm of every data point}}$ and assuming the transformed (logged) data to be $\boxed{\text{normally distributed}}$ — the sample mean and variance of $\ln(x)$ then estimate the lognormal's $\mu$ and $\sigma^2$ directly.
4. (d) Inverse transforms for all distributions. $\boxed{\text{False}}$. In principle, a generalized inverse of any valid CDF exists, but “can be found” in the practical sense of this course means a closed-form, analytically invertible expression $x=F^{-1}(U)$ — and several common distributions (Normal, Gamma with non-integer shape, Beta) have no closed-form CDF/inverse and must instead be generated by acceptance–rejection, convolution, or numerical/rational approximations.
5. (e) Poisson arrivals $\Rightarrow$ inter-arrival distribution. $\boxed{\text{Exponentially}}$ distributed — the standard Poisson-process duality between a Poisson count process and Exponential inter-event-time process.
6. (f) Poisson distributions model. $\boxed{\text{The number of independent, rare/random}}$ arrivals (events) occurring in a fixed interval, drawn from a $\boxed{\text{large (effectively infinite)}}$ population of potential arrivals, each with a small individual probability of arriving in that interval.
7. (g) Exponential as a special case. The exponential distribution is a special case of the $\boxed{\text{Gamma}}$ distribution (shape parameter $=1$) and of the $\boxed{\text{Weibull}}$ distribution (shape parameter $=1$).
8. (h) Inspections until first defect. $\boxed{\text{Geometric}}$ distribution (number of Bernoulli trials up to and including the first success/defect).
9. (i) Defectives in a batch drawn from a large population. With a large population and a fixed defective rate $p$, sampling without replacement is well-approximated by sampling with replacement, so the count is modelled by the $\boxed{\text{Binomial}(n,p)}$ distribution (the exact model, valid for any population size, is Hypergeometric; Binomial is the large-population approximation the question is pointing at).
10. (j) Triangular distribution. $\boxed{\text{A quantity for which only a minimum, a most-likely (mode), and a maximum value are known}}$ — the classic subjective/expert-opinion estimate used when there is too little data to fit a more rigorous distribution (e.g. early-stage activity-time estimates).