I have a linear path X(t)=a+b*t.

where a is fixed, and b is normally distributed with \mu and \sigma.

The time at which this path reaches F, is denoted by T_F.

Obviously TF=(F-a)/b.

I'm interested in the value of E[TF]. Intuitively, I'd say that E[TF]=(F-a)/E[b]; but some simulation analysis shows me I'm probably wrong.

Can someone help me out?