Can any of you help me?
I have some problem with this exercise of "Probability and Statistics" :
Calculate the probability density function (PDF) of $Z=X+Y$
where $Y$ is discrete random variable which is be equal to $-1,1$ with equal probability;
$X$ is standard Gaussian random variable independent from $Y$.
I know that the PDF of sum of two continuous independent variables is given by the convolution of the marginal PDF
$f_z(z)=f_x*f_y$
but if one on the two variable is discrete what should I do?
thanks all!
Best Answer
If one variable is discrete, you can use the Dirac delta function to preform the integration: $$P(X=x) = \frac{1}{2}\delta(x-1) + \frac{1}{2}\delta(x+1)$$ Down-votes from purist mathematicians in 3.. 2.. 1..