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Because <math>X, Y</math> are independent jointly distribute Poisson random variable.<br> | Because <math>X, Y</math> are independent jointly distribute Poisson random variable.<br> | ||
<math>P_{X+Y}(x,y)=P_X(x)\dot P_Y(y)</math><br> | <math>P_{X+Y}(x,y)=P_X(x)\dot P_Y(y)</math><br> | ||
| − | Such that <math>P_Z(z)=\sum_{x=0}^{z} e^(-\lambda)\dfrac{\lambda^x}{x!}e^{-\mu}\dfrac{\mu^{(z-x)}}{(z-x)!}=\dfrac{e^{-(\lambda+\mu)}}{z!}\sum_{x=0}^{z} \begin{pmatrix} z \\ x \end{pmatrix} \lambda^x\mu^{(z-x)}</math> | + | Such that <math>P_Z(z)=\sum_{x=0}^{z} e^(-\lambda)\dfrac{\lambda^x}{x!}e^{-\mu}\dfrac{\mu^{(z-x)}}{(z-x)!} |
| + | =\dfrac{e^{-(\lambda+\mu)}}{z!}\sum_{x=0}^{z} \begin{pmatrix} z \\ x \end{pmatrix} \lambda^x\mu^{(z-x)} | ||
| + | =e^{-(\lambda+\mu)}\dfrac{(\lambda+\mu)^z}{z!}</math><br> | ||
---- | ---- | ||
[[ECE-QE_CS1-2016|Back to QE CS question 1, August 2016]] | [[ECE-QE_CS1-2016|Back to QE CS question 1, August 2016]] | ||
[[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]] | [[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]] | ||
Revision as of 23:31, 18 February 2019
Communication Signal (CS)
Question 1: Random Variable
August 2016 Problem 3
Solution
a)
Because $ X, Y $ are independent jointly distribute Poisson random variable.
$ P_{X+Y}(x,y)=P_X(x)\dot P_Y(y) $
Such that $ P_Z(z)=\sum_{x=0}^{z} e^(-\lambda)\dfrac{\lambda^x}{x!}e^{-\mu}\dfrac{\mu^{(z-x)}}{(z-x)!} =\dfrac{e^{-(\lambda+\mu)}}{z!}\sum_{x=0}^{z} \begin{pmatrix} z \\ x \end{pmatrix} \lambda^x\mu^{(z-x)} =e^{-(\lambda+\mu)}\dfrac{(\lambda+\mu)^z}{z!} $
