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=Example. Addition of two jointly distributed Gaussian random variables= | =Example. Addition of two jointly distributed Gaussian random variables= | ||
| − | Let <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> be two jointly distributed Gaussian random variables. Assume <math>\mathbf{X}</math> has mean <math>\mu_{\mathbf{X}}</math> and variance <math>\sigma_{\mathbf{X}}^{2} , \mathbf{Y}</math> has mean <math>\mu_{\mathbf{Y}}</math> and variance <math>\sigma_{\mathbf{Y}}^{2}</math> , and that the correlation coefficient between <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> is <math>r</math> . Define a new random variable <math>\mathbf{Z}=\mathbf{X}+\mathbf{Y}</math> . | + | Let <math class="inline">\mathbf{X}</math> and <math class="inline">\mathbf{Y}</math> be two jointly distributed Gaussian random variables. Assume <math class="inline">\mathbf{X}</math> has mean <math class="inline">\mu_{\mathbf{X}}</math> and variance <math class="inline">\sigma_{\mathbf{X}}^{2} , \mathbf{Y}</math> has mean <math class="inline">\mu_{\mathbf{Y}}</math> and variance <math class="inline">\sigma_{\mathbf{Y}}^{2}</math> , and that the correlation coefficient between <math class="inline">\mathbf{X}</math> and <math class="inline">\mathbf{Y}</math> is <math class="inline">r</math> . Define a new random variable <math class="inline">\mathbf{Z}=\mathbf{X}+\mathbf{Y}</math> . |
(a) | (a) | ||
| − | Show that <math>\mathbf{Z}</math> is a Gaussian random variable. | + | Show that <math class="inline">\mathbf{Z}</math> is a Gaussian random variable. |
| − | If <math>\mathbf{Z}</math> is a Guassian random variable, then it has a characteristic function of the form | + | If <math class="inline">\mathbf{Z}</math> is a Guassian random variable, then it has a characteristic function of the form |
| − | <math>\Phi_{\mathbf{Z}}\left(\omega\right)=e^{i\mu_{\mathbf{Z}}\omega}e^{-\frac{1}{2}\sigma_{\mathbf{Z}}^{2}\omega^{2}}.</math> | + | <math class="inline">\Phi_{\mathbf{Z}}\left(\omega\right)=e^{i\mu_{\mathbf{Z}}\omega}e^{-\frac{1}{2}\sigma_{\mathbf{Z}}^{2}\omega^{2}}.</math> |
| − | <math>\Phi_{\mathbf{Z}}\left(\omega\right)</math> | + | <math class="inline">\Phi_{\mathbf{Z}}\left(\omega\right)</math> |
| − | where <math>\Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)</math> is the joint characteristic function of <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> , defined as | + | where <math class="inline">\Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)</math> is the joint characteristic function of <math class="inline">\mathbf{X}</math> and <math class="inline">\mathbf{Y}</math> , defined as |
| − | <math>\Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)=E\left[e^{i\left(\mathbf{\omega_{1}X}+\omega_{2}\mathbf{Y}\right)}\right].</math> | + | <math class="inline">\Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)=E\left[e^{i\left(\mathbf{\omega_{1}X}+\omega_{2}\mathbf{Y}\right)}\right].</math> |
| − | Now because <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> are jointly Gaussian with the given parameters, we know that | + | Now because <math class="inline">\mathbf{X}</math> and <math class="inline">\mathbf{Y}</math> are jointly Gaussian with the given parameters, we know that |
| − | <math>\Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)=e^{i\left(\mu_{X}\omega_{1}+\mu_{Y}\omega_{2}\right)}e^{-\frac{1}{2}\left(\sigma_{X}^{2}\omega_{1}^{2}+2r\sigma_{X}\sigma_{Y}\omega_{1}\omega_{2}+\sigma_{Y}^{2}\omega_{2}^{2}\right)}.</math> | + | <math class="inline">\Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)=e^{i\left(\mu_{X}\omega_{1}+\mu_{Y}\omega_{2}\right)}e^{-\frac{1}{2}\left(\sigma_{X}^{2}\omega_{1}^{2}+2r\sigma_{X}\sigma_{Y}\omega_{1}\omega_{2}+\sigma_{Y}^{2}\omega_{2}^{2}\right)}.</math> |
Thus, | Thus, | ||
| − | <math>\Phi_{\mathbf{Z}}\left(\omega\right)=\Phi_{\mathbf{XY}}\left(\omega,\omega\right)=e^{i\left(\mu_{X}\omega+\mu_{Y}\omega\right)}e^{-\frac{1}{2}\left(\sigma_{X}^{2}\omega^{2}+2r\sigma_{X}\sigma_{Y}\omega^{2}+\sigma_{Y}^{2}\omega^{2}\right)}</math><math>=e^{i\left(\mu_{X}+\mu_{Y}\right)\omega}e^{-\frac{1}{2}\left(\sigma_{X}^{2}+2r\sigma_{X}\sigma_{Y}+\sigma_{Y}^{2}\right)\omega^{2}}=e^{i\mu_{Z}\omega}e^{-\frac{1}{2}\sigma_{Z}^{2}\omega^{2}}</math> | + | <math class="inline">\Phi_{\mathbf{Z}}\left(\omega\right)=\Phi_{\mathbf{XY}}\left(\omega,\omega\right)=e^{i\left(\mu_{X}\omega+\mu_{Y}\omega\right)}e^{-\frac{1}{2}\left(\sigma_{X}^{2}\omega^{2}+2r\sigma_{X}\sigma_{Y}\omega^{2}+\sigma_{Y}^{2}\omega^{2}\right)}</math><math class="inline">=e^{i\left(\mu_{X}+\mu_{Y}\right)\omega}e^{-\frac{1}{2}\left(\sigma_{X}^{2}+2r\sigma_{X}\sigma_{Y}+\sigma_{Y}^{2}\right)\omega^{2}}=e^{i\mu_{Z}\omega}e^{-\frac{1}{2}\sigma_{Z}^{2}\omega^{2}}</math> |
| − | where <math>\mu_{Z}=\mu_{X}+\mu_{Y}</math> and <math>\sigma_{Z}^{2}=\sigma_{X}^{2}+2r\sigma_{X}\sigma_{Y}+\sigma_{Y}^{2}</math> . | + | where <math class="inline">\mu_{Z}=\mu_{X}+\mu_{Y}</math> and <math class="inline">\sigma_{Z}^{2}=\sigma_{X}^{2}+2r\sigma_{X}\sigma_{Y}+\sigma_{Y}^{2}</math> . |
| − | <math>\mathbf{Z}</math> is a Gaussian random variable with <math>E\left[\mathbf{Z}\right]=\mu_{X}+\mu_{Y} and Var\left[\mathbf{Z}\right]=\sigma_{X}^{2}+2r\sigma_{X}\sigma_{Y}+\sigma_{Y}^{2}</math> . | + | <math class="inline">\mathbf{Z}</math> is a Gaussian random variable with <math class="inline">E\left[\mathbf{Z}\right]=\mu_{X}+\mu_{Y} and Var\left[\mathbf{Z}\right]=\sigma_{X}^{2}+2r\sigma_{X}\sigma_{Y}+\sigma_{Y}^{2}</math> . |
(b) | (b) | ||
| − | Find the variance of <math>\mathbf{Z}</math> . | + | Find the variance of <math class="inline">\mathbf{Z}</math> . |
| − | As show in part (a) <math>Var\left[\mathbf{Z}\right]=\sigma_{\mathbf{X}}^{2}+2r\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}+\sigma_{\mathbf{Y}}^{2}</math> . | + | As show in part (a) <math class="inline">Var\left[\mathbf{Z}\right]=\sigma_{\mathbf{X}}^{2}+2r\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}+\sigma_{\mathbf{Y}}^{2}</math> . |
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Latest revision as of 12:00, 30 November 2010
Example. Addition of two jointly distributed Gaussian random variables
Let $ \mathbf{X} $ and $ \mathbf{Y} $ be two jointly distributed Gaussian random variables. Assume $ \mathbf{X} $ has mean $ \mu_{\mathbf{X}} $ and variance $ \sigma_{\mathbf{X}}^{2} , \mathbf{Y} $ has mean $ \mu_{\mathbf{Y}} $ and variance $ \sigma_{\mathbf{Y}}^{2} $ , and that the correlation coefficient between $ \mathbf{X} $ and $ \mathbf{Y} $ is $ r $ . Define a new random variable $ \mathbf{Z}=\mathbf{X}+\mathbf{Y} $ .
(a)
Show that $ \mathbf{Z} $ is a Gaussian random variable.
If $ \mathbf{Z} $ is a Guassian random variable, then it has a characteristic function of the form
$ \Phi_{\mathbf{Z}}\left(\omega\right)=e^{i\mu_{\mathbf{Z}}\omega}e^{-\frac{1}{2}\sigma_{\mathbf{Z}}^{2}\omega^{2}}. $
$ \Phi_{\mathbf{Z}}\left(\omega\right) $
where $ \Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right) $ is the joint characteristic function of $ \mathbf{X} $ and $ \mathbf{Y} $ , defined as
$ \Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)=E\left[e^{i\left(\mathbf{\omega_{1}X}+\omega_{2}\mathbf{Y}\right)}\right]. $
Now because $ \mathbf{X} $ and $ \mathbf{Y} $ are jointly Gaussian with the given parameters, we know that
$ \Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)=e^{i\left(\mu_{X}\omega_{1}+\mu_{Y}\omega_{2}\right)}e^{-\frac{1}{2}\left(\sigma_{X}^{2}\omega_{1}^{2}+2r\sigma_{X}\sigma_{Y}\omega_{1}\omega_{2}+\sigma_{Y}^{2}\omega_{2}^{2}\right)}. $
Thus,
$ \Phi_{\mathbf{Z}}\left(\omega\right)=\Phi_{\mathbf{XY}}\left(\omega,\omega\right)=e^{i\left(\mu_{X}\omega+\mu_{Y}\omega\right)}e^{-\frac{1}{2}\left(\sigma_{X}^{2}\omega^{2}+2r\sigma_{X}\sigma_{Y}\omega^{2}+\sigma_{Y}^{2}\omega^{2}\right)} $$ =e^{i\left(\mu_{X}+\mu_{Y}\right)\omega}e^{-\frac{1}{2}\left(\sigma_{X}^{2}+2r\sigma_{X}\sigma_{Y}+\sigma_{Y}^{2}\right)\omega^{2}}=e^{i\mu_{Z}\omega}e^{-\frac{1}{2}\sigma_{Z}^{2}\omega^{2}} $
where $ \mu_{Z}=\mu_{X}+\mu_{Y} $ and $ \sigma_{Z}^{2}=\sigma_{X}^{2}+2r\sigma_{X}\sigma_{Y}+\sigma_{Y}^{2} $ .
$ \mathbf{Z} $ is a Gaussian random variable with $ E\left[\mathbf{Z}\right]=\mu_{X}+\mu_{Y} and Var\left[\mathbf{Z}\right]=\sigma_{X}^{2}+2r\sigma_{X}\sigma_{Y}+\sigma_{Y}^{2} $ .
(b)
Find the variance of $ \mathbf{Z} $ .
As show in part (a) $ Var\left[\mathbf{Z}\right]=\sigma_{\mathbf{X}}^{2}+2r\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}+\sigma_{\mathbf{Y}}^{2} $ .
