还有问题（原：请教微积分或统计高手）

[cola]

大侠请留步，刚才少写了一样东西：

p is a constant, n is the variable, want to find
an expression of n in terms of p such that the above
summation is maximized

:thanks::thanks::thanks:

 

[gdntfrank]

A hintr[y=n]>Pr[y=n+1];
and: Pr[y=n]>Pr[y=n-1].
Solve the above two inequations, you should find the answer n interms of p.

 

[garion]

But Pr[y=n], Pr[y=n+1] and Pr[y=n-1] are sums of bernoulli, with different numbers of terms, how could the inequality be solved?

最初由 gdntfrank 发布
A hintr[y=n]>Pr[y=n+1];
and: Pr[y=n]>Pr[y=n-1].
Solve the above two inequations, you should find the answer n interms of p.

 

[garion]

It's not simply y=some value, but the sum shown above. There is a relationship between p and the n that will give the maximal value for the sum...

最初由 gdntfrank 发布
A hintr[y=n]>Pr[y=n+1];
and: Pr[y=n]>Pr[y=n-1].
Solve the above two inequations, you should find the answer n interms of p.

 

[wood]

just guess, not sure
binomial random variable? each trial is independent?
p is the probability of success?
then in 2n trials, the number of successes is y? then n = y/2p ?
then use y = ... sub in the binomial equation, and expand it??
again, just guess

 

[gdntfrank]

http://mathworld.wolfram.com/BinomialDistribution.html
Just some intuition. If p>0.5, then the larger N is, the larger Pr(N).
On the other side, if P<0.5, then N should be small.
Since each trial is independent and identical. The more time we try,
the result will more close to its mean Np. If p>0.5, then Np>N/2. When
N->infinity, Pr(N)->1.

 

[gdntfrank]

I guess, if p>=0.5, n=infinity.If p<0.5, n=1.

 

[garion]

i think there is some misunderstanding of the problem. a brief recap. this is to prove: for a bernoulli trial of size 2n, there is a maximum for the second half binomial tail (tail means the sum of bernoulli number from n+1 to 2n), and we want the expression for the n that gives this maximum in terms of bernoulli p.

if you plot it in matlab, you will see for instance, for p=0.47, n=8 will give the maximum tail.

最初由 gdntfrank 发布
I guess, if p>=0.5, n=infinity.If p<0.5, n=1.

 

[gdntfrank]

You mentionaed tail, that means p<0.5, otherwise it is not tail.
It also shows that if p>0.5, n should be infinity. The reason for
some p<0.5, n!=1, is beacuase of the discrete value of n. You
may have to solve the beta function.

 

[garion]

you are right, such maximum only exists when p<.5, otherwise, it monotonically increases. is there any way to work out the relationship without going into beta function?

最初由 gdntfrank 发布
You mentionaed tail, that means p<0.5, otherwise it is not tail.
It also shows that if p>0.5, n should be infinity. The reason for
some p<0.5, n!=1, is beacuase of the discrete value of n. You
may have to solve the beta function.