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# Thread: difference between P[min(X1, X2, X3) < 3] and P[max(X1, X2, X3) < 3]

1. ## difference between P[min(X1, X2, X3) < 3] and P[max(X1, X2, X3) < 3]

difference between P[min(X1, X2, X3) < 3] and P[max(X1, X2, X3) < 3]
So what exactly is the difference between P[min(X1, X2, X3) < 3] and P[max(X1, X2, X3) < 3]. I am trying to solve problems 4 and 5 from the ACTEX P/1 Study Manual 2010 Edition page 211. The way i see it is

P[min(X1, X2, X3) < 3] = P(X1 < 3)*P(X2 < 3)*P(X3 < 3)

which is the same as

P[max(X1, X2, X3) < 3] = P(X1 < 3)*P(X2 < 3)*P(X3 < 3)

The solution guide used the survival function for the min, which i dont understand why it couldnt be done my approach. the solution for the max was the way i have it written. it is a uniform distribtion by the way.  Reply With Quote

2. I don't have the book, but I can see what's happening: Your first statement is wrong because in words you are saying "The probability that the smallest of the three random variables is less than 3 is equal to the probability that all three random variables are less than 3."

I can make a counterexample: if the values are 2.5, 4, and 5, then the minimum is less than 3, even though the other two aren't.

You should find an equation that translates to "The probability that the smallest of the three random variables is less than 3 is equal to the probability that not all three random variables are greater than 3" which is logically true. This will lead you to the book solution. Hope this helps!  Reply With Quote