# Questions regarding an experiment containing a joint continuous and discrete random variable

For the following experiment:

A random number $$X$$ is chosen uniformly from $$[0, 1]$$. Then a sequence $$Y_1,Y_2\dots Y_i$$ of random numbers is chosen independently and uniformly from $$[0, 1]$$. The game ends the first time that $$Y_i > X$$.

If I define $$Z$$ as a discrete (countably infinite) random variable $$\{1,2,3...\}$$ that represents the number of turns before the game ends (inclusive of the last turn), $$P(Z=z|X=x)=(1-x)x^{z-1}$$.

1. The marginal probability mass function for $$Z$$ is: $$p_Z(z)=\int_{0}^{1}P(Z=z|X=x)f_X(x)dx=\int_{0}^{1}(1-x)x^{z-1}dx$$?

2. Is $$E(Z|X=x) = \frac{1}{1-x}$$, and hence $$E(Z) = \int_{0}^{1}E(Z|X=x)f_X(x)dx=\int_{0}^{1}\frac{1}{1-x}dx$$ which diverges?

3. How is $$f_{X,Z}(x,z)$$ defined, given that it is neither continuous nor discrete?

• 1. and 2. look correct. For 3. to get the z component of the density function you need to use delta functions at each value of z. – herb steinberg May 27 at 3:00

The joint distribution-mass distribution function of $$\ X\$$ and $$\ Z\$$ is given by $$P\left(X\le x, Z=z\right) =\cases{ 0 & for x<0 \\ \int_\limits{0}^x \left(1-y\right)y^{z-1}dy=\frac{x^z}{z}-\frac{x^{z+1}}{z+1} & for 0\le x<1 \\ \frac{1}{z(z+1)} &for 1\le x }\ ,$$ and their joint density-mass distribution function $$\ f_{X,Z}\$$ (assuming that's what you meant by this expression) is obtained by differentiating this with respect to $$\ x\$$: $$f_{X,Z}\left(x,z\right)=\cases{ 0& for x<0\\ x^z-x^{z+1} & for 0\le x<1 \\ 0 & for 1\le x }$$
Calculating $$\ E\left(Z\right)\$$ directly: $$\sum_\limits{z=1}^\infty zp_Z\left(z\right)=\sum_\limits{z=1}^\infty z\left( \frac{1}{z(z+1)}\right)=\sum_\limits{z=1}^\infty\frac{1}{z+1}$$ confirms your conclusion that it diverges (to $$+\infty$$).