# Sum of Log-normal and exponential random variables

I have two random variables X and Y. X is lognormally with pdf f(x) and CDF F(X), whereas Y is exponentially distributed with pdf g(y) and CDF G(y). In my application, I need to calculate CDF H(z) of Z = X+Y. I know one standard way that is mentioned widely in the books is convolution of X & Y i.e. \begin{align}H(z) = \int_{-\infty}^{\infty} F(z-y)g(y)dy \end{align} Are there any assumptions involves while evaluating this integral. Because in another application I tried to convolute gamma and exponentially distributed RVs, and depending on the value of parameters for distributions sometimes the result of integration was an imaginary number. This happened because in those cases scale parameter of gamma distribution was negative. So, I would like to know that is it mathematically possible to convolute X & Y as defined above.

• The bounds for $z$ are 0 and $\infty$. But I don´t think that a closed form can be obtained. – callculus May 31 at 5:33
• Yes you are absolutely right. Infact limits for integration are 0 to a for H(a) as F(a-y) and g(y) are defined for positive values. In my research problem, I am solving an optimization problem thus, I know value of 'x' and 'y' as well, so all I am left with is to evaluate the above integral H(z<=a), and I am using trapezoidal method to evaluate this definite integral, as closed form doesn't exists. My main concern here is that can I convolute X and Y as they have different distribution functions. Are there any assumptions that need to be satisfied? – Aditya Malik May 31 at 14:35