How Do You Calculate the Conditional PDF for a Given Union of Events?

[shespuzzling]

Hi,

This is a homework problem that I'm having a very hard time with. We are given that f(x)=e^(-x) for X greater/equal to zero. The question is to find f(x|A) where A is the union of the events (1 less/equal x, and x greater than 10). I can't figure out how to go about doing this...I thought of taking the complement of A and solving the conditional probability for x between 1 and 10, but then if I take 1 minus that, I don't think that f(x|A) will have an area less than 1, and it will also be negative at points. Any help is greatly appreciated.

Thank you.

 

[mathman]

f(x|A)=f(x)/∫_Af(u)du, for xεA
f(x|A)=0, otherwise

 

[shespuzzling]

Thanks for your help! But in this case, since A is less than 1 and greater than 10, what would the limits of the integral e? Also, what is f(u) in this case?

 

[mathman]

The integral just gets split up into two parts (0,1) and (10,inf).

Your second question makes me wonder about what level of math you are at. f is the function you are given. For the purpose of expressing the integral, u is just the dummy variable for integration - any other letter would do. I deliberately did not use x, since x is being used as the explicit variable for the expression.