# Stochastic Processes - Probability Theory

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?

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 $$P(X_t\ge 0|t = \frac{2}{3})=\frac{3}{4}$$.