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I have tried to solve the following statements:

a) The number of leaves on a fully grown oak tree – Normal

b) The age at which people's hair turns grey – Normal

c) The number of hairs on the heads of 20-year olds – Normal

d) The number of people who have been hit by lightning x times - Power Law

e) The number of miles driven before your car needs a new transmission - Poisson

f) The number of runs a batter will get, per cricket over – Poisson

g) The number of leopard spots per square foot of leopard skin – Normal

h) The number of people with exactly x pennies sitting in drawers – Binomial

i) The number of apps on people's cell phones – Normal

j) The daily attendance in Skiena's data science course - Normal

I am not sure about the following statements, please give me your feedback.

(a) The beauty contestants at the Miss Universe contestant.

(b) The gross of movies produced by Hollywood studios.

(c) The birth weight of babies.

(d) The price of art works at auction.

(e) The amount of snow New York will receive on Christmas.

(f) The number of teams that will win x games in a given football season.

(g) The lifespans of famous people.

(h) The daily price of gold over a given year.

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  • $\begingroup$ What is The Quant Shop? $\endgroup$
    – Dave
    Oct 17, 2020 at 7:45
  • $\begingroup$ @Dave It is just a term used in this question. Please ignore it. $\endgroup$ Oct 17, 2020 at 7:47
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    $\begingroup$ The questions may make more sense given some context. At best, you have been told about a small number of named distributions and are asked to guess which might be most appropriate. Even so, the question can't be answered without knowing which distributions are candidates. Several of your answers suggest "normal" for distributions that are in principle discrete. Answers of "no idea but show me some data and I will give you some comments" or "is there any reason why that should follow a named distribution?" are presumably not expected, but are likely from anyone experienced. $\endgroup$
    – Nick Cox
    Oct 17, 2020 at 10:43

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Rather than directly answer each point, I think it is better to explain how to think about these questions.

I would disagree with even your answer to h). Just because the question states the number of pennies does not mean it has anything to do with flipping coins. For each scenario you should consider the distributions, and what they represent. For instance, can we have a binomial distribution with n and p to adequately represent the number of coins someone will have? I would suggest not, because if you set an 'n', then you eliminate the possibility that someone has n+1 coins, effectively saying it has 0 probability.

Additionally, can the number of leaves on a tree truly be normally distributed? I can sample 187.231 from the normal distribution, but can we have that many leaves. In practice it would be okay to use a normal distribution to model discrete values as it is indeed "close enough", but here I would say it is more appropriate to model it with the discrete poisson distribution, (which is asymptotically normal anyway).

In summary, have a look at the properties of the distributions, consider not only their shape, but what values can be sampled from them, and then use that information to decide which model to use for each problem.

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