Probability Density of Sum of Random Variables

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Homework Statement


Suppose X and Y are independent random variables with X following a uniform distribution on (0,1) and Y exponentially distributed with parameter [itex]\lambda = 1[/itex]. Find the density for Z = X + Y. Sketch the density and verify it integrates to 1.



Homework Equations


If Z = X + Y, and X and Y are independent,

[tex]f_z(z) = \int_{-\infty}^{\infty}f_Y(z-x)f_X(x)\,dx[/tex]


The Attempt at a Solution


I am trying to catch up on stats. I think I screwed up somewhere in this problem:

If X ~ Unif(0,1), then [itex]f_X(x)[/itex] should just be 1 on (0,1) right? Also if the parameter of the exponential distribution is 1, then [itex]f_Y(z-x) = e^{-(z-x)}?[/itex]

But then the expression for [itex]f_z(z)[/itex] doesn't seem to make much sense. Note I have no idea what I am doing.
 

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