求两个指数分别相加服从的分布
求两个指数分别相加服从的分布
\(X\)~\(\epsilon(\lambda)\) \(e^{-\lambda t}\)
\(Y\)~\(\epsilon(\mu)\) \(e^{-\mu t}\)
\[F(X+Y)=P(X+Y>t)=\iint\limits_{X+Y>t}f(x,y)dxdy\\=\int_{0}^{t}{f_x(x)}dx\int_{t-x}^{\infty}{f_y(y)}dy + \int_{t}^{\infty}{f_x(x)}dx\int_{0}^{\infty}{f_y(y)}dy\\=\int_{0}^{t}{\lambda e^{-\lambda x}}dx\int_{t-x}^{\infty}{\mu e^{-\mu y}}dy + \int_{t}^{\infty}{\lambda e^{-\lambda x}}dx\int_{0}^{\infty}{\infty}{\mu e^{-\mu y}}dy\\=\frac{\mu}{\lambda - \mu}e^{-\lambda t} - \frac{\lambda}{\lambda - \mu}e^{-\mu t}
\]
\[F^`(X+Y)=f(x,y)\\=\frac{\mu\lambda}{ \mu - \lambda }e^{-\lambda t} - \frac{\lambda\mu}{\mu - \lambda }e^{-\mu t}\\=\frac{\mu\lambda}{\mu - \lambda}(e^{-\lambda t}-e^{-\mu t})
\]
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