Conditional Probability: Converting CDF to PDF for Independent Random Variables

In summary, the conversation discusses the process of turning a conditional cumulative distribution function (cdf) into a conditional probability density function (pdf) when the random variables are independent. The individual states that they have only seen this explained using the definition of independence and questions a step in the provided solution. They also mention that the inequality g(x,y) < z cannot always be rewritten as x < h(y,z) and provide an example. They suggest considering a fixed y and taking the integral with respect to x for a more recognizable form.
  • #1
joshthekid
46
1
Basically I am wondering how you deal with a conditional cdf and turning that into a conditional pdf when the random variables are independent. I know that f(X|Y) =f(X)f(Y)/f(Y)=f(X)

I tried to derive this in a nice attached laTex document but it does not seem right to me.

Note(this is for a homework problem but this is only a derivation I am trying to use to solve it so I decided to post it here because it is not a textbook problem)
 

Attachments

  • ECE514hw5.pdf
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  • #2
I have only seen this explained as
##f(X|Y) = \frac{f(X \cap Y)}{f(Y)} ## where ##f(X \cap Y) = f(X)f(Y)## by the definition of independence.
In your work, it seems like in part (6) you were taking the integral with respect to y, where you should be considering a fixed y and taking the integral with respect to x.
I have not put pen to paper, but it looks like that could get you something in a more recognizable form.
 
  • #3
The inequality [itex] g(x,y) < z [/itex] can't necessarily be rewritten in the form [itex] x < h(y,z) [/itex].

For example, the solution [itex] x^2 + y < z [/itex] might require that [itex] x [/itex] be in an interval of the form [itex] -a < x < a [/itex] rather than in an interval of the form [itex] x < a [/itex].
 

1. What is conditional probability?

Conditional probability is a mathematical concept that measures the likelihood of an event occurring given that another event has already occurred. It is denoted as P(A|B), which reads as the probability of event A given that event B has occurred.

2. How is conditional probability calculated?

Conditional probability is calculated by dividing the probability of the joint occurrence of both events A and B by the probability of event B. This can be expressed as P(A|B) = P(A and B) / P(B).

3. What is the difference between conditional probability and regular probability?

The main difference between conditional probability and regular probability is that conditional probability takes into account the occurrence of another event, while regular probability does not. In other words, conditional probability looks at the probability of an event happening given that another event has already occurred, while regular probability looks at the probability of an event happening without taking into consideration any other events.

4. How is conditional probability used in real life?

Conditional probability has many real-life applications, such as predicting the likelihood of a disease given certain risk factors, determining the probability of a customer purchasing a product based on their demographic, and predicting the likelihood of a sports team winning a game based on their performance in previous games.

5. What is the importance of understanding conditional probability?

Understanding conditional probability is important for making informed decisions and predictions based on existing data. It allows us to consider the influence of one event on another, and to more accurately calculate the likelihood of a certain outcome. It is also a fundamental concept in fields such as statistics, machine learning, and data analysis.

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