Two independent random variables, change the line integral plus joint distribution function, outage probability, there are two random variables

Language 2020-03-06 12:34:29 views: null

1, the random variable hsubject to complex Gaussian distribution, namely h\sim CN(0,\lambda_{1}), the X = \ h green \ green ^ {2}exponential distribution, that is X \ sim E (\ lambda_ {1}).

XDensity function can be written as

f(x)=\left\{\begin{matrix} \lambda_{1}e^{-\lambda_{1}x}, x\geq 0 \\ 0,\ \ \ \ \ \ \ \ x<0 \end{matrix}\right.，

XThe distribution function can be written as

\left\{\begin{matrix} P(X\geq x)=e^{-\lambda_{1}x}\\ P(X\leq x)=1-e^{-\lambda_{1}x} \end{matrix}\right.。

2, during a communication outage probability requirements, you may encounter Pr(X\ge aY+b)form, wherein X \ yes E (\ {1} lambda_), Y \ yes E (\ {2} lambda_)two exponential distribution random variable, a, ba constant, is calculated as follows:

\begin{aligned} Pr(X\ge aY+b) &=E_{Y}[F_{X}(ay+b)]\\ &= \int_{0}^{\infty}f_{y}(t_{1})Pr(X\ge at_{1}+b)dt_{1}\\ &=\int_{0}^{\infty}f_{y}(t_{1})\int_{at_{1}+b}^{\infty}f_{x}(t_{2})dt_{2}dt_{1}\\ &=\int_{0}^{\infty}f_{y}(t_{1})e^{-\lambda_{1}(at_{1}+b)}dt_{1}\\ &=\int_{0}^{\infty}\lambda_{2}e^{-\lambda_{2}t_{1}}e^{-\lambda_{1}(at_{1}+b)}dt_{1}\\ &=\lambda_{2}e^{-\lambda_{1}b}\int_{0}^{\infty}e^{(-\lambda_{2}-a\lambda_{1})t_{1}}dt_{1}\\ &=\lambda_{2}e^{-\lambda_{1}b}/(\lambda_{2}+a\lambda_1) \\ \end{aligned}

 

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