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How do you find the joint CDF from a joint pdf?
To find the joint CDF for x>0 and y>0, we need to integrate the joint PDF: FXY(x,y)=∫y−∞∫x−∞fXY(u,v)dudv=∫y0∫x0fXY(u,v)dudv=∫min(y,1)0∫min(x,1)0(u+32v2)dudv.
How do you find the CDF when given a pdf?
Relationship between PDF and CDF for a Continuous Random Variable
- By definition, the cdf is found by integrating the pdf: F(x)=x∫−∞f(t)dt.
- By the Fundamental Theorem of Calculus, the pdf can be found by differentiating the cdf: f(x)=ddx[F(x)]
How do you find the marginal CDF from a joint CDF?
If we know the joint CDF of X and Y, we can find the marginal CDFs, FX(x) and FY(y). Specifically, for any x∈R, we have FXY(x,∞)=P(X≤x,Y≤∞)=P(X≤x)=FX(x). Here, by FXY(x,∞), we mean limy→∞FXY(x,y). Similarly, for any y∈R, we have FY(y)=FXY(∞,y).
How to define joint pdf and joint CDF of X and Y?
Suppose we have a discrete random variable X and a continuous random variable Y. I am trying to understand how one defines/ find the joint PDF and joint CDF of X and Y.
Can you define joint pdf of a discrete and continuous variable?
If both the random variables were discrete (continuous) then we could have found the joint PMF (joint PDF). But since X and Y are discrete and continuous, respectively, can we define “hybrid” joint PDF or “hybrid” joint PMF?
Is there a simple rule to differentiate this and get the PDF?
We can get the joint pdf by differentiating the joint cdf, Pr (X ≤ x, Y ≤ y) with respect to x and y. However, sometimes it’s easier to find Pr (X ≥ x, Y ≥ y). Notice that taking the complement doesn’t give the joint CDF, so we can’t just differentiate and flip signs. Is there still some simple rule to differentiate this and get the pdf?
Is the PDF the same as the CDF?
The answer is yes, and the PDF is exactly what you say it is. If you don’t want to use measure theory, then you have to take what you say as the definition of the PDF in this setting. This is why everything in elementary probability has two versions, one for discrete and one for continuous.