Bounds for the mean of the minimum of binomial random variables

[soroush1358]

Dear Friends,
I want to find an upper and lower bound for the expected value of the minimum of independent binomial random variables. What paper/book do you suggest for this problem?

In other words, I need to find bounds for E(min(X1,X2,...,Xn)), where Xi 's are independent random variables with binomial distribution.

Thanks in advance.

 

[EnumaElish]

Why do you need bounds, have you thought of deriving/computing the expected value directly?

 

[soroush1358]

There is not any close formula for the cdf of binomial distribution. Hence, it seems that the minimum can not be evaluated theoretically. As a result of this, I prefer to find some upper and lower bounds for it.

 

[EnumaElish]

There is not any close formula for the cdf of binomial distribution.

You mean, other than this floor-sum expression: http://en.wikipedia.org/wiki/Binomial_distribution#Cumulative_distribution_function ?

 

[blue_raver22]

Dear colleague,

Thank you for your question. Finding bounds for the expected value of the minimum of independent binomial random variables is a common problem in statistics and probability theory. There are several papers and books that discuss this topic, but I would recommend starting with the book "Probability and Random Processes" by Geoffrey Grimmett and David Stirzaker. Chapter 3 of this book covers the topic of order statistics, which includes the minimum of independent random variables. Additionally, the paper "Bounds for the Distribution of the Minimum of Independent Random Variables" by David Siegmund and Michael Marcus may also be helpful in your research. I hope this helps and good luck with your work!