Can the Addition Formula for PDFs Also Apply to CDFs?

[nikozm]

Hello,

i was wondering if the additon formula for PDFs holds also for CDFs.

Particularly, when an X event or a Y event may occur then the total PDF= PDF [X] + PDF [Y].

The same goes for CDFs ? i.e. total CDF = CDF [X] +CDF [Y] ?


Any help would be useful.
Thanks in advance

 

[Stephen Tashi]

Particularly, when an X event or a Y event may occur then the total PDF= PDF [X] + PDF [Y].

What do you mean by the "total" PDF? - what random variable is the "total" PDF a PDF for?

Are you assuming X and Y are mutually exclusive events?

 

[nikozm]

yes, X and Y are mutually exclusive events. Now, let g = X given a constraint, while g = Y given another constraint.
Then the unconditional (total) PDF of g is well-known to be: PDF [g] = PDF [X] + PDF [Y].

My question, however, is simply if the addition rule for probability density function also holds for CDFs..

Thanks

 

[Stephen Tashi]

yes, X and Y are mutually exclusive events. Now, let g = X given a constraint, while g = Y given another constraint.
Then the unconditional (total) PDF of g is well-known to be: PDF [g] = PDF [X] + PDF [Y].

Can you rephrase that using standard terminology?

"PDFs" are something associated with "random variables". "Events" have "probabilities", not "PDFs".
"Events" can be "mutually exclusive". I've never encountered the term "mutually exclusive" applied to "random variables". How would that be defined?

 

[nikozm]

ok. I m interested in continious random variables, so i refer to PDFs and CDFs in the following:

X and Y are independent random variables. Now, let g = X given a constraint, while g = Y given another constraint.

Then the unconditional (total) PDF of g is well-known to be: PDF [g] = PDF [X] + PDF [Y].
My question, however, is simply if the addition rule for probability density function also holds for CDFs..

Thanks

 

[Stephen Tashi]

I'm sorry, I still don't understand.

let g = X given a constraint, while g = Y given another constraint.

Then the unconditional (total) PDF of g

If g is has a PDF, I assume g must be a random variable. So is "g = X" a constraint that is defined by the equality of two random variables?

Can you give a link or specific example?

 

[nikozm]

ok i will try to be more specific.

Let g = a*X when a> r and g = a*Y when a < r. Assume a, X, Y are independent non-negative random variables and r is non-negative constant value.
Then it holds that: PDF [g] = PDF[a*X, given that a > r] + PDF[a*Y, given that a < r].

I am just wondering if the above formula also holds if we substitute PDFs with CDFs..

 

[Stephen Tashi]

Then it holds that: PDF [g] = PDF[a*X, given that a > r] + PDF[a*Y, given that a < r].

If A and B are events then
Probability(A) = Probability(A and B) + Probability (A and not-B).
Probability(A) = Probability(A given B) Probability(B) + Probability(A given not-B) Probability(not-B)

So I don't see why the result isn't

PDF[g](s) = PDF[a*X | a > r)](s) Probability( a > r) + PDF[a*Y| a < r)](s) Probability(a < r)

 

[nikozm]

Can you just tell me if the addition rule for probabilities stands also for CDFs in general ?

 

[Stephen Tashi]

Can you just tell me if the addition rule for probabilities stands also for CDFs in general ?

First we have to get straight what you mean by "the addition rule" as applied to PDFs. If f and g are PDFs then f + g won't be a PDF since the integral of f+g over the real line will be 2 instead of 1.

 

[nikozm]

http://en.wikipedia.org/wiki/Probability_axioms#More_consequences

 

[Stephen Tashi]

Are you're referring to $P(A \cup B) = P(A) + P(B) - P(A \cap B)$?

I don't understand how you intend to apply that a PDF or CDF. If you are referring to a "well known" result about PDFs then please give a link to this result about PDFs.

If you are asking about a situation involving two different random variables X,Y then you are dealing with a joint density function h(x,y) of two variables. Some authors define the "joint cumulative distribution" for the joint density. Are you asking a question about the joint cumulative distribution?