# 为什么我等的公交车总是迟到？

#### 数学证明

\begin{align} P\{X_{N(t)+1} \geq x\} &= \int_0^{\infty} P\{X_{N(t)+1} \geq x | S_{N(t)} = s \}dF_{S_{N(t)}}(s)\\ &= \int_0^{\infty} P\{X_{N(t)+1} \geq x | X_{N(t) + 1} > t – s \}dF_{S_{N(t)}}(s) \\ &= \int_0^{\infty} \frac{P\{X_{N(t)+1} \geq x, X_{N(t) + 1} > t – s \}}{P\{X_{N(t) + 1} > t – s \}}dF_{S_{N(t)}}(s) \\ &= \int_0^{\infty} \frac{1 – F(max\{x, t-s\})}{1 – F(t-s)}dF_{S_{N(t)}}(s) \\ &= \int_0^{\infty} min\{\frac{1 – F(x)}{1 – F(t-s)}, \frac{1 – F(t-s)}{1 – F(t-s)}\}dF_{S_{N(t)}}(s) \\ &= \int_0^{\infty} min\{\frac{1 – F(x)}{1 – F(t-s)}, 1\}dF_{S_{N(t)}}(s) \\ &\geq \int_0^{\infty} 1 – F(x)dF_{S_{N(t)}}(s) \\ &= \bar{F}(x) \end{align}

$$F(x) = 1 – e^{-\lambda x}$$

\begin{align} P\{X_{N(t)+1} \geq x\} &= \int_0^{\infty} \frac{e^{-\lambda x} }{e^{-\lambda(t-s)}}dF_{S_{N(t)}}(s) \\ &= \int_0^{t-x} dF_{S_{N(t)}}(s) + \int_{t-x}^t e^{-\lambda x} dm(s) \\ &= – e^{-\lambda t} + (1 + x/\lambda)e^{-\lambda x} \\ & = – e^{-\lambda t} > \bar{F}(x) \end{align}

#### 经济学的困境

Reference:

Ross, S. M., Kelly, J. J., Sullivan, R. J., Perry, W. J., Mercer, D., Davis, R. M., … & Bristow, V. L. (1996). Stochastic processes (Vol. 2). New York: Wiley.