# What is the pdf of sum of log-normal and normal distribution?

The question goes like this: $Z = X+Y$; where

$X$ is Log-normal Random variable with parameters - $\mu = 0 \quad \sigma^2= 1$,

$Y$ is Gaussian Random variable with $\mu= 0\quad \sigma^2= 1$

What is the pdf of $Z$? I know it will be the convolution of $X$ and $Y$. However, I am unable to solve it. Is it even solvable?

PS: $X$ and $Y$ are independent.

• Are we assuming that $X$ and $Y$ are independent? – Math1000 May 28 '16 at 4:04
• I'm afraid you will have difficulty finding an analytical solution given that the characteristic function $$\varphi_X(t) = \sum_{n=0}^\infty \frac{(it)^n}{n!}e^{\frac 12 n^2}$$ does not converge. – Math1000 May 28 '16 at 4:25
• Yes, the random variables $X$ and $Y$ are independent. – skt9 May 28 '16 at 16:01

By saying convolution, you mean the two random variables $X$ and $Y$ are independent and the joint probability density function of them can be represented as the convolution of their pdfs.

Let $X$ be the log-normal random variable, and $Y$ the normal one, the pdf's of which are as below in the figure.

The probability density function of $Z=X+Y$ cannot be represented in closed form, but the numerical results of the pdf $f_Z(x)$ can be evaluated by numerical integral.

$$f_Z(x)=\int_{-\infty}^{+\infty}f_X(t)\cdot f_Y(x-t){\rm{d}}t=\int_{t=0}^{+\infty}\dfrac{e^{-\tfrac{1}{2}\left( (t-x)^2+{\ln^2t}\right)}}{2\pi t}{\rm{d}}t$$

So a better way to answer this question might be to visualize them as below:

• Thank you. Is it possible to know the expression for $f_Z$$(x)$ by any means? Can I know the tool used for performing numerical integration and getting the graph above? – skt9 May 28 '16 at 16:03
• hi @skt9, the analytical expression for $f_Z(x)$ has been given as above. There are a lot of special functions which have no closed forms (expression by elementary functions) but can be numerically obtained or visualized easily. -- A powerful tool in calculating the numerical integral and visualizing the profile is Wolfram Mathematica, which is also what I was using in getting the graph here. – LCFactorization May 29 '16 at 0:32