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Let $X_1,X_2,...,X_n$ be a fixed number of Bernoulli random variables. My problem is to find a distribution for $Y$ such that for some function $f$, we have $Y=f(X_1,X_2,...,X_n)$. There are two candidate functions to use, $max$ or $avg$. I have no idea if an average function would work here or not but I think it'd give me a meaningful result.

I have looked into similar problems and mostly found the cases where $X_i$ are continuous random variables. Any hint on this problem is highly appreciated.

Thanks!

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  • $\begingroup$ This sounds like homework. Also, it's not clear what you mean by two candidate functions. Both max and avg, along with dozens of other functions f will give you a distribution. Are there conditions on Y that you've forgotten to include? $\endgroup$ – Gschneider Apr 5 '12 at 13:01
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    $\begingroup$ It sounds like you're just being asked to derive the distribution of the average of a bunch of bernoulli trials (think about a scaled binomial distribution) and the maximum. There are a number of threads about deriving the distribution of the maximum of a set of random variables, so I'd suggest searching for those $\endgroup$ – Macro Apr 5 '12 at 13:07
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    $\begingroup$ @user10405 Those were polite and justified comments; there is no need to react emotionally. $\endgroup$ – user88 Apr 6 '12 at 7:45
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User10405, I didn't mean to offend you by the homework comment, but proving that the sum of independent Bernoullis is a Binomial is exactly a homework problem I assigned the statistics class (Mathematical Statistics 1) I teach recently. Apologies for the "useless prejudice."

I'm confused, Macro's comment seemed appropriate, yet:

Macro: This is not a bunch of Bernoulli trials and a binomial distribution does not apply.

But in the original statement:

Let $X_1,X_2,...,X_n$ be a fixed number of Bernoulli random variables

See http://en.wikipedia.org/wiki/Bernoulli_random_variable for verification that the sum of Bernoullis is indeed a Binomial, and since $avg$ is just the sum divided by $n$, it will be a scaled Binomial.

As far as $f=max$ goes, since each of the X 's is a Bernoulli, i.e. 0 or 1, the max also has to be 0 or 1. It will be 0 only if all of the X 's are 0. So treat it as a Bernoulli($p∗$) where $p∗=1−P(X_1=X_2=...=X_n=0)$ .

An assumption of independence among the $X$'s makes $p∗$ much easier to calculate.

I get the feeling that this answer still won't be what you're looking for, but without more clarification, I don't see how we can help further.

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    $\begingroup$ Yes this is how I was thinking of the problem. If the candidate function is the average, then $Y$ having a (scaled) binomial distribution would have the desired property and you've given such a distribution for $max$. Either a crucial detail is missing from the problem description, or our point is not being fully understood. $\endgroup$ – Macro Apr 6 '12 at 1:35

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