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I have the following equation

[tex]\mathbf{v}(t)=\mathbf{P}(t)\mathbf{d}+\mathbf{w}(t)[/tex]

where v(t) is a 2-by-1 vector, P(t) is 2-by-2N matrix, d is a 2N-by-1 vector, and w(t) is an 2-by-1 Gaussian process vector where each element is of zero mean and variance N0. What is the probability distribution function (p.d.f.) of v(t) given that P(t) and d? I know it is Gaussian with mean P(t)d and covariance matrix N0 I_2, but I am not sure how to write it. Is the following right:

[tex]p(\mathbf{v}(t)\Big|\mathbf{P}(t),\mathbf{d})=A \exp\left(-\frac{1}{N_0}\int_{-\infty}^{\infty}\Big\|\mathbf{v}(t)-\mathbf{P}(t)\mathbf{d}\Big\|^2\,dt\right)[/tex]

where A is some constant, since I am concerned only for the exponential argument. I appreciate your help

Thanks

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# Continous Time Gaussian Distribution

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