Finding Conditional Probabilities/Requirements on lambda

Question

I'm looking to get some insight into a moderately challenging conditional probability problem:

Consider a sequence of random variables $X_1$,$X_2$...,$X_n$ which each take the values 0 and 1. Assume that Pr($X_j$ = 1) = 1 - Pr($X_{j-1}$= 0) = $\phi$, $\hspace{5mm}$j =1,...,n

where 0 < $\phi$ < 1 and that

$\hspace{10mm}$Pr($X_j$ = 1|$X_{j-1}$=1) = $\lambda$, $\hspace{5mm}$j = 2,...,n.

(a) Find Pr($X_j$ = 0|$X_{j-1}$=1), Pr($X_j$ = 1|$X_{j-1}$=0),Pr($X_j$ = 0|$X_{j-1}$=0). (b) Find the requirements on $\lambda$ so that this describes a valid probability distribution for $X_1$,$X_2$...,$X_n$.

So for part (a) I have: 1-$\lambda$, $\phi$, and 1-$\phi$, respectively. I do not know how to begin part (b).

Answer 1

For $u=0$ and $u=1$, let $q_u=P[X_j=u\mid X_{j-1}=u]$, then $1-q_u=P[X_j=1-u\mid X_{j-1}=u]$. Since $P[X_j=1]=P[X_{j-1}]=\phi$, one has $\phi=q_1\phi+(1-q_0)(1-\phi)$. Since $q_1=\lambda$, this yields $q_0=1-(1-\lambda)\phi/(1-\phi)$. But note that $q_0$ ought to be a transition probability, hence one should have $0\leqslant q_0\leqslant1$. Thus:

The requirements for $(\lambda,\phi)$ to describe a bona fide distribution are that $$ 0\leqslant\phi\leqslant1,\quad 0\leqslant\lambda\leqslant1,\quad (2-\lambda)\phi\leqslant1. $$

Note that the i.i.d. case is when $\lambda=\phi$, in which case the set of conditions reduces to $0\leqslant\phi\leqslant1$.