Let $\{X_t ,t\ge0\} $ be Poisson process with parameter $\lambda $ and arrival time $T_1,T_2,\cdots$ , Find $Var(T_2-t|X_t=1)$
My try: First I want to find the conditional density of $T_2-t$ given $X_t=1$, so I consider $$P(T_2-t\lt s|X_t=1)=\frac{P(T_2-t\lt s,X_t=1)}{P(X_t=1)}=\frac{P(T_2\lt t+ s,X_t=1)}{P(X_t=1)}=\frac{P(1 \text{ arrivals in } [0,t],2 \text{ arrivals in }(t,t+s])}{P(X_t=1)}=\frac{P(X_t=1,X_{t+s}-X_t=1)}{P(X_t=1)}=\frac{P(X_t=1)P(X_{t+s}-X_t=1)}{P(X_t=1)}=P(X_{t+s}-X_t=1)=P(X_s=1)=e^{-\lambda s}\frac{(\lambda s)^1}{1!} $$
What's wrong with this process?