Gaussian is a continuous p.d.f and Poisson is discrete p.d.f. Let x be a random variable with Gaussian distribution and y be r.v with Poisson. Can a variable z=xy exist, if so then with what p.d.f? If not is there a way to make normal distribution discrete and do so?
Can two random variables let one be Gaussian and another be Poisson distributed be multiplied?
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1 Answers
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Yes $Z$ exists, and it is continuous except for the fact that it has a point mass at 0. In fact you can show that its CDF satisfies $\mathbb{P}[Z\leq z]=\sum_{n=0}^\infty\mathbb{P}[nX\leq z]\mathbb{P}[Y=n]$, which is differentiable away from $z=0$. If you want to make a Normal distribution discrete, the easiest way to do this is by using the floor or ceiling function, as they round the Normal to make it an integer.
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0Can you please also tell me the resource(Books, e-material) where I can study this theory in detail? I need to model a process involving such P.D.F. I do not think that normal undergraduate Probability books include this, but not sure. – 2017-02-22
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0There's probably nothing with this exact process, but this is just another example of what you study under the umbrella of joint distributions. Questions like "What is the distribution of $X+Y$, $XY$, $\frac{X}{Y}$" given the distribution of $X$ and $Y$ are elementary problems found in whatever undergraduate textbook you can get your hands on. – 2017-02-22