Joint probability distribution of random variables in Bernoulli process

Question

I am interested in studying a Bernoulli process $\{X_i | i \in \mathbb{N} \}$ with parameter $p$. Consider the following random variables:

• $S = \min\{i \ | \ X_i = 1\}$

• $I = \begin{cases} 1, & \text{if $X_1 = X_2$} \\ 0, & \text{otherwise} \end{cases}$

• $F = \min\{i \ | \ X_i = 0\}$

I believe that $S$ and $F$ are geometric random variables with parameters $p$ and $1-p$ respectively. I am trying to find the expectation of $I$. By first principles, this is:

$$ E(I) = \sum_{i=1}^{2}x_ip_i(x_i) = p(1) $$

How do I calculate this value? Will I need the probability mass function of $I$ or is it simpler?

The second question concerns the joint distribution of $(S,I,F)$. Some triples of numbers here are excluded, namely the ones where $S = F$ as it is not possible to have the first success and failure occur at the same time. Moreover, the triples $(1,0,0)$ and $(0,0,1)$ are the only valid ones when $I = 0$.

So the question really becomes, what are the permissible triples when $I = 1$? They should be of the form $(n, 1, m)$ where $n<m$ or $m < n$ but how can we be more specific?

And finally, after having determined the permissible triplets, how do we assign probability to them? Any suggestions on how to approach this will be greatly appreciated!

Answer 1

I believe that $S$ and $F$ are geometric random variables with parameters $p$ and $1−p$ respectively.

Yes, they are such, and their support is $\{0,1,2,\ldots\}$; to be clear on which kind of geometric distribution we're talking about.

I am trying to find the expectation of $I$.

Hint: $I$ is valued as either $1$ or $0$ due to a single decision's success : "Does $X_1=X_2$ ?"

What kind of random variable has this behaviour?

The second question concerns the joint distribution of $(S,I,F)$ . Some triples of numbers here are excluded, namely the ones where $S=F$ as it is not possible to have the first success and failure occur at the same time.

Yes. You can go further. If $S=k$ for some $k\geq 1$, what possible values of $F$ are there, and vice versa. (That is, if the first 'success' is , say $5$, when was the first 'failure'?)

Moreover, the triples $(1,0,0)$ and $(0,0,1)$ are the only valid ones when $I=0$

Corresponding to events of $X_0=1, X_1=0, X_2=1$ and $X_0=0, X_1=1, X_2=0$.

$(0,0,2)$ is also possible; corresponding to: $X_0=1, X_1=1, X_2=0$. And symmetrically, $(2,0,0)$ corresponds to $X_0=0, X_1=0, X_2=1$

So the question really becomes, what are the permissible triples when $I=1$.

$S=0, I=1$ requires $\{X_0=1, X_1=X_2\}$, so ...