Consider two independent random variables $X$ and $Y$, where $X$ is uniformly distributed on the interval $[0,1]$ and $Y$ is uniformly distributed on the set $\{0,1\}$. Thus, the cdfs are given by
$F_X(x) =
\begin{cases}
0 & x <0\\
x & 0 \leq x <1\\
1 & else
\end{cases}$
and
$F_Y(y) =
\begin{cases}
0 & y <0\\
1/2 & 0 \leq y <1\\
1 & else.
\end{cases}$
Consider the random variable $Z = (X,Y)$ with cdf is given by
$F_Z(x,y) =
\begin{cases}
0 & y<0\\
F_X(x)/2& 0 \leq y < 1\\
F_X(x) & else.
\end{cases}$
Now, what I'm interested in is a pdf of $Z$. Is
$
f_Z(x,y) =
\begin{cases}
0 & y<0\\
f_X(x)/2& 0 \leq y < 1\\
f_X(x) & else.
\end{cases}
$
the pdf of $Z$? More generally, I'm interested in the joint pdf of independent random variables, one of which is continuous and the others (possibly more than one) are discrete. If correct, can the above be applied in this case?
Thank you for your help.
Best Answer
No. If one of the variables is discrete and the other continuous, they can't have a common density [neither with respect to the Lebesgue-measure, nor the counting measure].