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I have a statistical model, $Y_i=Bf_i+W_i$, where $i=1,\ldots,N$

$B\thicksim\mathcal{N}(0,\sigma_B^2)$

$W_i\thicksim\mathcal{N}(0,\sigma_w^2)$

$B,W_0,\ldots,W_N$ are i.i.d. and $f_i$ is a known deterministic signal.

I need to write the likelihood function $p_{\mathbf{Y}}(\mathbf{y})$ for this model.

My initial thought was to write $\mathbf{y}$ as $\mathbf{y}\thicksim\mathcal{N}\left(\mathbf{0}_N,(\sigma_B^2+\sigma_w^2)\,\mathbf{I}_N\right)$, where $\mathbf{0}_N$ is a column vector of zeros with length $N$ and $\mathbf{I}_N$ is the identity matrix with dimenson $N$. But I think this would be incorrect unless $B$ was also a random process, $B_i\thicksim\mathcal{N}(0,\sigma_B^2)$, with $i=1,\ldots,N$.

If $B$ was a constant, $b$ then I could write (I think) $\mathbf{y}\thicksim\mathcal{N}\left(b\mathbf{1}_N,\sigma_w^2\,\mathbf{I}_N\right)$.

I am stuck trying to figure out how to complete the sum of a random variable to a random process. Would welcome suggestions/tips/pointers.

HVW
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1 Answers1

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Since $B, W_i$ are normal iid so is their sum $Y_i = f_iB + W_i$ with parameters $$ \Bbb E[Y_i] = f_i\cdot \Bbb E[B] + \Bbb E[W_i] $$ and $$ \Bbb V[Y_i] = f^2_i\Bbb V[B] + \Bbb V[W_i] $$

SBF
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  • From the helpful explanation immediately above, could I take the question one step further? What if $B$ has some correlation with $W_i$? Would it be correct to say that $\mathrm{var}(Y_i)=f_i^2\sigma_B^2+\sigma_w^2+2\rho(f_i\sigma_B+\sigma_w)$? – HVW Jul 10 '21 at 07:41