How Do You Calculate the PDF of Z=X/Y with X Uniform and Y Exponential?

[Kalinka35]

Homework Statement

Let X be uniformly distributed over [0, 1] and Y be exponentially distributed with λ=1. Assuming X and Y are independent, give the PDF of Z=X/Y.

Homework Equations

The Attempt at a Solution

I know the individual PDFs of X and Y. I was trying to use a geometric approach to find the CDF and then integrate but since Y is exponential the geometry doesn't really work. Are there any other methods I can use to get this? I haven't seen any examples resembling this.

Thanks.

 

[mighty2000]

To find the PDF of Z, we can use the transformation method. First, we need to find the CDF of Z, which can be done by using the following formula:

FZ(z) = P(Z ≤ z) = P(X/Y ≤ z) = P(X ≤ zY)

Since X and Y are independent, we can rewrite this as:

FZ(z) = P(X ≤ z)P(Y ≤ Y) = (z)(1-e^(-λY))

Next, we can differentiate the CDF to get the PDF of Z:

fZ(z) = d/dz FZ(z) = 1-e^(-λY)

Substituting in the value of Y as 1 (since λ=1), we get:

fZ(z) = 1-e^(-1)

Therefore, the PDF of Z is 1-e^(-1). I hope this helps! Let me know if you have any further questions or if you need clarification on any steps.