Can Conditional Probability Be Solved Generally with PDFs of Variables?

In summary, the conversation discusses using the CDF method for a random variable as a function of X. The goal is to convert P() expressions to X_CDF() expressions. The speaker also mentions needing to reformulate expressions and asks about the case of having P(A + B < y) with knowledge of A_PDF(a) and B_PDF(b).
  • #1
rabbed
243
3
Is it possible to solve something like this generally or does it depend on the pdf's of the variables?

P(x < f(y) | x > -f(y))
 
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  • #2
Your expression can be given as [itex]\frac{P(-f(y)<x<f(y))}{P(-f(y)<x<\infty)}[/itex].
 
  • #3
As a step in using the CDF method for a random variable as a function of X where i have X_PDF, I came from:

P(x > -f(y) AND x < f(y)) =
P(x > -f(y)) * P(x < f(y) | x > -f(y))

The aim is to convert the P()'s to X_CDF()'s.

Your answer led me back a step, which made me think that maybe P(x > -f(y) AND x < f(y)) might be expressed like (1-X_CDF(-f(y))) - X_CDF(f(y))

It seems to be correct for my case, so thank you :)
 
  • #4
By the way..

In the CDF method, I understand that I need to reformulate expressions to get something like P(X < y) which equals X_CDF(y) or P(X > y) which equals (1-X_CDF(y)), since I know the expression of X_PDF(x) = X_CDF'(x).

What if I have P(A + B < y), knowing A_PDF(a) and B_PDF(b)?
Would that require that I know AplusB_PDF(a,b) and some transformation from y to a and y to b?
 

1. What is a conditional probability?

A conditional probability is the likelihood of an event occurring given that another event has already occurred. It is denoted by P(A|B), where A is the event of interest and B is the condition.

2. How is a conditional probability calculated?

A conditional probability is calculated by dividing the probability of the joint occurrence of both events (P(A∩B)) by the probability of the condition (P(B)). In formula form, it is written as P(A|B) = P(A∩B) / P(B).

3. What is the relationship between conditional probability and independence?

If two events A and B are independent, then the conditional probability of A given B is equal to the probability of A. In other words, the occurrence of event B does not affect the likelihood of event A. However, if two events are not independent, then the conditional probability of A given B may be different from the probability of A.

4. How is a conditional probability used in real-life applications?

Conditional probability is used in a variety of fields, including medicine, finance, and weather forecasting. For example, in medicine, it can be used to determine the chance of a patient having a certain disease given their age and other risk factors. In finance, it can be used to calculate the probability of different investment outcomes based on market conditions. In weather forecasting, it can be used to predict the likelihood of a hurricane given certain meteorological conditions.

5. What is the difference between conditional probability and joint probability?

Conditional probability calculates the likelihood of an event given that another event has already occurred, while joint probability calculates the likelihood of two events occurring together. In other words, conditional probability is a subset of joint probability. Additionally, conditional probability uses the probability of the condition in its calculation, while joint probability uses the probability of both events occurring simultaneously.

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