The probability for the minimum of a series of random variables to be in a range.

Question

Given a value $A$ and a range around it $\left[\left(1-\epsilon\right)A,\left(1+\epsilon\right)A\right]$ and a random variable $\chi_i$. The probability for $\chi_i\geq\left(1-\epsilon\right)A$ is $Pr\left[\left(1-\epsilon\right)A\right]>p_1$ and the probability for $\chi_i\geq\left(1-\epsilon\right)A$ is $Pr\left[\left(1+\epsilon\right)A\right]{\leq}p_2$.

$Ex\left[\chi_i\right]=A$, $p_1=1-\gamma$, where $0<\gamma<\frac{1}{2}$ and I can control the value of $\gamma$, and $p_2$ was calculated using Markov's inequality.

What is the probability that the minimum value for $\chi_i$ from $m$ values is in the reange $\left[\left(1-\epsilon\right)A,\left(1+\epsilon\right)A\right]$?

I figured that in order to get this, I would need for at least one value to be less then $\left(1+\epsilon\right)A$ and all of them to be more than $\left(1-\epsilon\right)A$.

The probability for all the values to be more than $\left(1+\epsilon\right)A$ is $p_2^m$, so the probabilty for at least one value to be less than that is $1-p_2^m$.

The probability for all the values to be more than $\left(1-\epsilon\right)A$ is $p_1^m$, so the total probabilty should be $p_1^m\left(1-p_2^m\right)$.

Is this correct?

Answer 1

You want to calculate the probability that both of the following events happen:

Let event $X$ be that at least one variable is less than $(1+\epsilon)A$, I agree that the probability of this is $(1-p_2^m)$

Let event $Y$ be that every variable is greater than $(1-\epsilon)A$ I agree that the probability of this is $p_1^m$

If these two events where independent we would be done, this is unlikely however.

What we need is $P(Y)P(X|Y)$.