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Suppose $X$ is random variable with PDF $f(X)=2(x-1)$, $1 \le x \le 2$; $Y$ is a random variable with a triangle pdf with minimum at $2$, mode at $2.5$, and maximum at $3$.

Is it possible to define a random variable like $Z$ which is the summation of $X$ and $Y$? ($Z=X+Y$, $X$ and $Y$ are independent)

In this situation, what is the PDF of variable $Z$?

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    $\begingroup$ Is it homework? What do you mean by "medium"? Is mode ($c$ parameter for triangular distribution) equal to 2.5? $\endgroup$ – Tim Oct 2 '16 at 19:00
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If $X$ and $Y$ are independent random variables the PDF of $Z=X+Y$ is given by

$f_{z}(z)=\int_{-\infty}^\infty f_{x}(z-y)\, f_{y}(y)\, dy,$

here is a link with examples .

This operation is known as a convolution. You can calculate the integral directly or you can use Laplace Transforms or Fourier Transforms. I think these transforms are referred to as Moment Generating Functions in the Statistics literature.

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    $\begingroup$ Not quite (in relation to your last sentence) -- Laplace transforms are (apart from a sign in the exponent) MGFs but Fourier transforms (again, aside from a sign in the exponent) characteristic functions. $\endgroup$ – Glen_b -Reinstate Monica Oct 3 '16 at 4:21

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