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.

 

[blue_raver22]

I understand the frustration you are facing with this problem. Let's break it down step by step to find the conditional pdf.

First, we need to understand what the conditional pdf represents. It is the probability distribution of X given that A has occurred. In other words, it is the probability of X taking on a certain value, given that we know A has occurred.

Next, we need to find the probability of A occurring. We can do this by finding the area under the curve of f(x) between 1 and 10. This gives us the probability of A occurring, which we will denote as P(A).

Now, to find the conditional pdf, we need to divide the original pdf f(x) by P(A). This is because we want the probabilities to sum up to 1, and by dividing by P(A), we are essentially scaling the probabilities to account for the fact that A has occurred.

So, the conditional pdf, denoted as f(x|A), would be:

f(x|A) = (1/P(A)) * f(x) for x between 1 and 10

And for x less than or equal to 0 or greater than 10, f(x|A) would be 0 since we know A has not occurred.

I hope this helps you understand how to approach this problem. Remember, it's always important to understand the concept behind a problem before trying to solve it. Good luck!