Finding the PDF of Z = X₁ + X₂ + X₁X₂ for independent exponential variables

[EngWiPy]

Hello,

Suppose that:

$$Z=X_1+X_2+X_1X_2$$

where $$X_i$$ for i=1 and 2 are independent and identically distribuited exponential RVs.

can we find the PDF of Z?

Regards

 

[EnumaElish]

You should note that the event {Z = z} is equivalent to {X1 + X2 + X1 X2 = z} or {X1 = (z - X2)/(1 + X2)}. You can use this and use convolutions (example).

 

[EngWiPy]

You should note that the event {Z = z} is equivalent to {X1 + X2 + X1 X2 = z} or {X1 = (z - X2)/(1 + X2)}. You can use this and use convolutions (example).

Right, But I want to find the PDF directly, not from differentiating the CDF, if possible. Because these RVs are, actually, not exponentials, but I said so to simplify the problem statement. So I want to avoid the derivative operation, which complicates the whole stituation.

I say the following:

let $$W=X_1+X_2$$ and $$Y=X_1X_2$$, then $$Z=W+Y$$. But we need to evaluate joint PDF of W and Y. Is this approach in the right way?

 

[SW VandeCarr]

Hello,

Suppose that:

$$Z=X_1+X_2+X_1X_2$$

where $$X_i$$ for i=1 and 2 are independent and identically distribuited exponential RVs.

can we find the PDF of Z?

Regards

You may want to check the f distribution. The PDF is a bit complicated and I don't have Latex, but you can look it up.

 

[EnumaElish]

You can derive the pdf directly through convolution.

 

[EngWiPy]

You can derive the pdf directly through convolution.

If we assume that $$Y=W+Z$$ where $$W=X_1X_2$$ and $$Z=X_1+X_2$$, then we need to find the joint PDF $$f_{W,Z}(w,z)$$, which can be found using Jacobian transformation.

If we proceed using this, we have:

$$X_1=T_1^{-1}=\frac{W+Z-X_2}{1+X_2}$$ and $$X_2=T_2^{-1}=\frac{W+Z-X_1}{1+X_1}$$

Then

$$F_{W,Z}(w,z)=f_{X_1,X_2}(x_1=T_1^{-1},x_2=T_2^{-1})|J|$$

where $$|J|$$ is the magnitude of the Jacobian which will be zero in this case!

Is here anything wrong I did?

Regards

 

[EnumaElish]

You can write X1 as (z - X2)/(1 + X2). Then study the wiki example with normal distribution. How is that example similar to your problem?