Need help with setting up 2 double integrals

[cse63146]

Homework Statement

I'm trying to find the CDF of e^-y.

The cdf of any f(x,y) is $$\int^x_{-\infty} \int^y_{-\infty} f(u,v) dv du$$

where u and v are just dummy variables. I need to consider 2 cases: 1) 0<x<y and 2) 0 < y < x. In other words I need to do 2 double integrals using the general set up of a CDF.

Homework Equations

The Attempt at a Solution

$$\int^x_0\int^y_x e^{-v} dv du$$ This case is when 0<x<y

$$\int^x_0\int^x_y e^{-v} dv du$$ This case is when 0<y<x

Did I set these up correctly? Because a friend of mine had 2 different answers, and I made a mistake somewhere. Thank you.

 

[mighty2000]

Thank you for your question regarding finding the CDF of e^-y. It looks like you have set up the integrals correctly for the two cases. However, I would suggest using a different approach to find the CDF.

Since we are dealing with a continuous random variable (e^-y), the CDF can be found by integrating the probability density function (PDF). In this case, the PDF is e^-y and the integration limits will be from -∞ to x.

So, the CDF for e^-y can be written as F(x) = ∫e^-y dy from -∞ to x. This can be simplified to 1-e^-x, as shown below:

F(x) = ∫e^-y dy from -∞ to x

= e^-x - e^-(-∞)

= e^-x - 0

= 1-e^-x

I hope this helps! Let me know if you have any further questions.Scientist