Let $N$ be a discrete random variable which takes values in [0, …, M], M > 0, with known PDF $P(N=n)$. Let also the continuous random variable $Z = \sum_{i=1}^{N}X_{i}$ as the sum of i.i.d. $X_{i}$ continuous variables with known and same PDF $f_{X_{i}}(x_{i}) = f_{X}(x), i=1\ldots,N$. The joint PDF of $N$ and $Z$ is $f_{Z,N}(z,n) = f_{Z|N}(z|n) P(N=n)$. What is the $f_{Y}(y) = f_{NZ}(nz)$ (i.e., Y = NZ)?
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Probability – What is the PDF of a Product of a Continuous and Discrete Random Variable?
probabilityprobability theory
Best Answer
First of all, a continuous and a discrete random variable don't have a joint PDF, i.e. their joint distribution is not absolutely continuous with respect to $2$-dimensional Lebesgue measure.
The cdf of $Y = NZ$ is $F_Y(y) = P(NZ \le y) = \sum_{n=0}^m P(N=n) P(nZ \le y | N=n)$. Note that if $N=0$ has nonzero probability, so will $Y=0$, so $Y$ doesn't have a pdf either (it has a mixed distribution with a discrete part and a continuous part).