Please help!! Thank you, Q;Which of these random variables have a hyper geometric distribution? Explain why.

Question

1. Which of these random variables have a hyper geometric distribution? Explain why.

a) the number of clubs dealt from a deck

b) the number of attempts before rolling a six with a die

c) the number of 3s produced by a random number generator

d) the number of defective screws in a random sample of 20 taken from a production line that has a 2% defect rate

e) the number of male names on a page selected at random from a telephone book

f) the number of left-handed people in a group selected from the general population



g) the number of left-handed people selected from a group comprised equally of left-handed and right-handed people

Answer 1

Hi Lia K.,

Your lucky day. I'm only helping with ne problem tonight -- yours.

Hypergeometric distribution refers to a situation resulting from a very specific type of selection: a certain number of items are drawn without replacement from a larger population.

So let's see which of your situations fits that definition!

"the number of clubs dealt from a deck" -- here, you have to assume that a subset of the deck has been chosen -- otherwise, you would simply get them all back, no problem! So, let's say, you select 13 cards (as in the standard 4-person dealing of all cards). Then you consider that value, the number of clubs you were dealt. Were these cards chosen without replacement? Yes, they were, they stayed in your pile until you looked at them after the deal. Unless you are cheating big time!

"the number of attempts before rolling a six with a die" -- this is a different kind of problem. You are not considering a result of a fixed number of operations; rather, you are operating until something happens. So that is a different distribution -- a Poisson, or something.

"the number of threes produced by a random number generator" -- here, you must supply some reasonable limits. You would not, for example, assume that the generator operated for an infinite time! But for the first 100 values generated, perhaps, how many 3's were there? And then you have to further assume that the random number generator only produced integers from 0 to 10, say. Otherwise, if it was producing real numbers, over ANY interval including 3, the probability of EXACTLY getting a 3 is zero! Not you have defined the problem properly, and you can confidently say that each result is independently produced, so therefore it is NOT a hypergeometric distribution. If you "got " a 3, there is still an infinite number of 3's waiting to be "gotten", the probability of getting another one did not diminish just because you "got" one. So this is like sampling WITH replacement.

"defective screws" - no, they are each an independent object when you selected.

"male names" - no, each name is independently male or female (or anyway, that used to be the case)

"left-handed, general population" -- no, assume the general population is infinite in size. Unless word gets out that the lefties aren't ever coming back -- that might skew your chances of capturing -- er, selecting -- another one

"lefties from a specified composition group" -- yes! Because your chances of getting the next leftie are significantly changed in this small group.

Hope this helped you. Remember, getting each"positive" result MUST affect the chances of the next "positive" result, for it to be a "hypergeometric distribution".

--Cheers, --Mr. d.