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It's an exercise from the online book Bayes Rules! An Introduction to Bayesian Modeling with R. Thus the book has no chapter with the right answers and so far I've not found a designated place to discuss its exercises, let's do the discussion here:)

Exercise 2.3 (Binomial practice) For each variable Y below, determine whether Y is Binomial. If yes, use notation to specify this model and its parameters. If not, explain why the Binomial model is not appropriate for Y.

a. At a certain hospital, an average of 6 babies are born each hour. Let Y be the number of babies born between 9 a.m. and 10 a.m. tomorrow.

I think that the Binomial model is not appropriate for Y here. First of all, we don't have any fixed number of trials n, relative to which we could have represented the number of babies born as the number of successes. Secondly, for the Binomial model we'd need the probability of baby birth to be defined. Obviously, the average of 6 babies born each hour cannot serve as either the fixed number of trials or the probability of baby birth.

b. Tulips planted in fall have a 90% chance of blooming in spring. You plant 27 tulips this year. Let Y be the number that bloom.

Y is Binomial here. Bloom of a tulip doesn't depend on blooms of the other tulips. We have the fixed number of trials n=27 in boundaries of which we count the number that bloom Y. The probability of bloom $\pi$=0.9 is defined as common for each bloom. So we can specify the Binomial model here as $$Y|0.9 \sim Bin(27, 0.9)$$

c. Each time they try out for the television show Ru Paul’s Drag Race, Alaska has a 17% probability of succeeding. Let Y be the number of times Alaska has to try out until they’re successful.

I think that the Binomial model is not appropriate for Y here. The Binomial model could have worked if Y was the number of successes.

d. Y is the amount of time that Henry is late to your lunch date.

I think that the Binomial model is not appropriate for Y here. It's impossible in this case to represent the amount of time as the number of successes out of some fixed number of trials.

e. Y is the probability that your friends will throw you a surprise birthday party even though you said you hate being the center of attention and just want to go out to eat.

I think that the Binomial model is not appropriate for Y here. Y must be a natural number.

f. You invite 60 people to your “π day” party, none of whom know each other, and each of whom has an 80% chance of showing up. Let Y be the total number of guests at your party.

Y is Binomial in this case. Thus the invited people do not know each other then the fact that somebody shows up could not influence whether other people come over or not. Also the probability of showing up $\pi=0.8$ does not change while events are occurring. The number of trials n=60 is fixed. So we can specify the Binomial model here as $$Y|0.8 \sim Bin(60, 0.8)$$

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All of your answers are correct and your reasoning for each is also correct.

As an exercise, for those random variables that are not well-modeled by a binomial distribution, what would be a plausible choice of parametric distribution?

For example, I think a Poisson distribution would be acceptable for (a), specifically $$Y \sim \operatorname{Poisson}(\lambda = 6).$$

What could you use for (c), (d), and (e)? Of course, these are not required to be modeled in a particular way, but what distributions could you use?

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  • $\begingroup$ Thank you very much for challenging me! As a stats newbie I need some time to work this out. For example, I've not known what a Poisson distribution yet:) But I've already started googling:) $\endgroup$
    – Raibek
    Oct 17, 2021 at 10:13
  • $\begingroup$ So far I think that the Poisson distribution is only applicable for the case of (a). That's because we have $\lambda=6$ and the interval of continuum which happens to be "each hour" of time is also specified. $\endgroup$
    – Raibek
    Oct 19, 2021 at 9:34
  • $\begingroup$ The case (c) definitely falls into the Geometric distribution with the probability of succeeding exponentially decreasing with each new trial. $\endgroup$
    – Raibek
    Oct 19, 2021 at 12:56

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