Stochastic Processes - Probability Theory

Question

Define the stochastic process $X_t = 2A + 3Bt$ where $P(A=2)=P(A=-2)=P(B=2)=P(B=-2)=\frac{1}{2}$. Find $P(X_t\geq0 | t)$.

I do know that we are supposed to start by doing the joint probability table first, but I am not sure where to go from there. Can someone please help me with this?

Answer 1

Assumption: $A$ and $B$ are independent.

Since both $A$ and $B$ are symmetrically distributed around $0$, then $P(X_t\gt 0)=P(X_t\lt 0)$

To complete the analysis we need $P(X_t=0)$

Here we need to distinguish a special case: $t=\frac{2}{3}$ In that case if $A$ and $B$ have opposite signs $X_t=0$ and $P(X_t=0) =\frac{1}{2}$

However in general $X_t$ cannot $=0$.

Summary $P(X_t\ge 0|t\ne \frac{2}{3})=\frac{1}{2}$, and $P(X_t\ge 0|t = \frac{2}{3})=\frac{3}{4}$.