Show $ \mathbb{E}(T_1\mid N_1=0)=1+\frac{1}{\lambda}$ for a Poisson process

Question

For a Poisson process with rate $\lambda$, show $ \mathbb{E}(T_1\mid N_1=0) = 1 + \frac{1}{\lambda}$.

attempt

We know $T_1 \sim \operatorname{Exp}(\lambda)$.

The solution I'm looking at says to use the memoryless property. I don't see how it fits. All I see is that $\{N_1 =0\}=\{T_1>1\}$. So I am stuck.

Answer 1

Hint: $E(T_1\mid N_1=0)=\int_0^\infty P(T_1 >t\mid N_1=0) \, dt$. For $t<1$ the integrand is $0$ so this becomes $\int_1^\infty P(T_1 >t\mid T_1>1) \, dt$. Now use the memoryless property.

Answer 2

$$ \operatorname E(T_1\mid T_1>1) = 1 + \operatorname E(T_1) \text{ by memorylessness.} $$ $$ = 1 + \frac 1 \lambda. $$