 # Eays Quant Question

The random variable X(i): number of heads if you throw a coin ten times, with i=1…200 The random variable which is the mean of the 200 randomvariables A) follows a binomial distribution and can be approximated by a normal distribution B) is biased although it follows a lognormaldistribution C) can be used to test if the coin is a fair coin, is a discrete random variable and follows approx. a normal distribution D) is with a probability of exactly 100 % lower then 10

I would say A… Best, TheChad

not A

it has to be C

Well, at a first glance I thought it was A, but then I changed my answer to C. Milos

maratikus my quant. I saw you on the board and knew you would be joining this question. Thought that Joey would beat you by a quicker response. But may be some others want to join the discussion, why A, B and D are wrong.

Well, the answer is sure not under B and C. And the concept of the question imply that A is also wrong. Maybe. Milos

Distribution of the means are approximately normal regardless of the distribution of the population, which in this case is a uniform discrete distribution. A is wrong because the population is not binomial (it’s uniform discrete), same thing with B. D is not wrong because the expected number of the throws is 5, with some probability > 0 of some outliers hitting straight heads.

Dreary Wrote: > D is not wrong because the expected number of the > throws is 5, with some probability > 0 of some > outliers hitting straight heads. D is wrong because P(mean of simulation = 10) = P(all 2,000 times head came up) = (1/2)^(2000) > 0

Exactly, and that’s not 100% zero is it?

@dreary - The population follows a binomial distribution. - and what do you mean by this D… sentence? D is wrong because: there is a probability that all 2000 realisations (<=>mean=10) are “head”. So the mentioned probability can’t be exactly 100 %.

But if you say it is binomial, wouldn’t that make A correct? Regarding D, there is a probability of more than zero that the mean will be 10. This can happen if all 200 trials show all heads everytime. No?

OK, A is not correct.

I hate answers that say “exactly” or “definitely”. I know there are exceptions to the rule, but they are usually wrong.

cfaisok Wrote: ------------------------------------------------------- > The random variable X(i): number of heads if you > throw a coin ten times, with i=1…200 > > The random variable which is the mean of the 200 > randomvariables > > A) follows a binomial distribution and can be > approximated by a normal distribution Nope If the X(i)'s are binomial their mean can’t be binomial. Only normals, poissons, and mixtures of those have that property. > B) is biased although it follows a > lognormaldistribution Way out there > C) can be used to test if the coin is a fair coin, If x-bar is no near 5, the coin ain’t fair. > is a discrete random variable Yep discrete rv’s => mean is discrete > and follows approx. > a normal distribution The CLT > D) is with a probability of exactly 100 % lower > then 10 P(X-bar) = 10 is mightily small but happens w/ prob 1/2^2000 for a fair coin.

@JoeyD I love your explanations, couldn’t say it in a better way, exept one thing. "If the X(i)'s are binomial their mean can’t be binomial. Only normals, poissons, and mixtures of those have that property. " The mean of poisson distributed rvs don’t have to be poisson distributed.

How about a multiple of Poisson? And there are some other ones too now that I think about it - gammas, negative binomials.

Pls, give an example Joey for a multiple of poisson where the mean is a new poisson distribution. negative binomials are from the type comparable to binomials and as you said in A --> the distribution type (not binom) of the mean is not the same as of the “population” (binom). —> I know that there is a repro property for the sum of poissons, but for the mean?

All members and candidates stop aside please, make way please… let these two stats heads slug it out. Kepp distance, lest you get hurt.

@dreary: nice that you are warning the others May be maratikus can join this discussion, I was impressed by his bright mind in a thread a few month ago…