ECE PhD QE CNSIP 2015 Problem1.3 - Rhea

Solution 1

Let $ \lambda = \frac{1}{\mu} $, then $ E(X)=E(Y)=\frac{1}{\lambda} $.

$ \phi_{X+Y}=E[e^{it(X+Y)}]=\int_{X}\int_{Y}e^{it(X+Y)}p(x,y)dxdy $

As X and Y are independent

$ \phi_{X+Y}=\int_{X}\int_{Y}e^{it(x+y)}p(x)p(y)dxdy = \int_{X}e^{itx}p(x)dx\int_{Y}e^{ity}p(y)dy=\phi_{X}\phi_{Y} $

$ \phi_{X}=E[e^{itX}]=\int_{-\infty}^{\infty}e^{itx}\lambda e^{-\lambda x} dx \\ = \lambda \int_{-\infty}^{\infty}e^{-(\lambda -iu)x} dx = -\frac{\lambda}{\lambda-iu}e^{-(\lambda-iu)x}|_0^\infty\\ =\frac{\lambda}{\lambda-iu} $

So $ \phi_{X+Y}=E[e^{it(X+Y)}]=\phi_{X}\phi_{Y} =( \frac{\lambda}{\lambda-iu})^2 $

Solution 2

$ \phi_X(w)=E[e^{iwX}]=\int_0^{+\infty}e^{iwX}\frac{1}{\mu}e^{-\frac{x}{\mu}}dx=e^{X(iw-\frac{1}{\mu})}\frac{1}{\mu}\frac{1}{iw-\frac{1}{\mu}}|_0^{+\infty}\\ = 0 - \frac{1}{\mu}\cdot\frac{1}{iw-\frac{1}{\mu}}=\frac{1}{1-iw\mu}\\ $

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