To solve this problem, we need to compute the probability density function (PDF) of $Z = X + Y$, given the independent random variables $X$ and $Y$ with their respective PDFs:

$f_X(x) = 5e^{-5x}, \quad x > 0$ $f_Y(y) = 2e^{-2y}, \quad y > 0$

The PDF of $Z$, $f_Z(z)$, can be determined using the convolution formula for independent random variables:

$f_Z(z) = \int_{-\infty}^\infty f_X(x) f_Y(z - x) \, dx$

Given $X > 0$ and $Y > 0$, $Z = X + Y$ will only have support for $z > 0$. We will compute the convolution only over the valid range $x > 0$ and $z - x > 0$, which implies $0 < x < z$. Thus, the limits of integration are adjusted accordingly:

$f_Z(z) = \int_0^z f_X(x) f_Y(z - x) \, dx$

Substitute the given PDFs:

$f_Z(z) = \int_0^z \left( 5e^{-5x} \right) \left( 2e^{-2(z - x)} \right) dx$

Simplify the expression:

$f_Z(z) = \int_0^z 10 e^{-5x} e^{-2z + 2x} \, dx$ $f_Z(z) = 10 e^{-2z} \int_0^z e^{-3x} \, dx$

The integral of $e^{-3x}$ is:

$\int e^{-3x} \, dx = -\frac{1}{3} e^{-3x}$

Evaluate the integral over the limits $0$ to $z$:

$\int_0^z e^{-3x} \, dx = \left[ -\frac{1}{3} e^{-3x} \right]_0^z = -\frac{1}{3} e^{-3z} + \frac{1}{3}$

Substitute this result into the expression for $f_Z(z)$:

$f_Z(z) = 10 e^{-2z} \left( \frac{1}{3} - \frac{1}{3} e^{-3z} \right)$ $f_Z(z) = \frac{10}{3} e^{-2z} \left( 1 - e^{-3z} \right)$

Thus, the PDF of $Z$ is:

$f_Z(z) = \frac{10}{3} e^{-2z} \left( 1 - e^{-3z} \right), \quad z > 0$

Final Answer:

$f_Z(z) = \frac{10}{3} e^{-2z} \left( 1 - e^{-3z} \right), \quad z > 0$

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Would you like a step-by-step explanation of the convolution process or clarification on any part?

Related Questions:

1. How does the convolution formula work for dependent random variables?
2. What happens to the PDF if $X$ and $Y$ are not independent?
3. How do the exponential distributions affect the shape of the resulting PDF?
4. What is the mean and variance of $Z = X + Y$?
5. Can this method be extended to other sums of random variables?

Tip:

When solving convolution problems, always carefully consider the limits of integration based on the support of the random variables.