Question

I do not know its the right forum to ask question, however i will appreciate if someone can help me.

I have two processes and each have a distinct random Variable , let say X1 & X2, each Random Variable is from a uniform distribution with [0,1], then how random.nextdouble() can help me to identify the variation between the probabilities of these two random variables. I need this variation because i want to find the probability of minimum of the two random variables.

Can I say that its too simple and I should run the program for 100000 or more times twice and then count the minimum value from two iterations? If so, then how can I map this result with probabilities of two random variables i.e. X1 & X2?? Like what is the criteria to say that the first time I ran the program was for X1 and 2nd time for X2.

Answer 1

The probability of a single variable under uniform distribution to be under d is P(X<=dx) = d (assuming in range [0,1]).

Thus, the probability of it to be more then d is P(X>=d) = (1-d).

The probability of 2 random variables to be above d is P(X>=d AND Y>=d) = P(X>=d)*P(Y>=d) = (1-d)^2

Thus, the probability that one of X or Y to be under d is p = 1-(1-d)^2, and this means that the probability of the minimum to be under d is the same: p = 1 - (1-d)^2.

If you are looking for the probability density function, you can just find the derivitive of the probability:

f(x) = d/dx P(x) = d/dx 1 - (1-x)^2 = 
     = d/dx (1 - 1 + 2x - x^2) = 
     = d/dx (2x - x^2) = 2 - 2x