1
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.