A discrete-valued parameter with the prior pdf $$ p(x) = \sum_{i=1}^{2}p_i\delta(x-1) $$ is measured with the additive noise $w \sim \mathcal{N}(0,\sigma^2)$ $$ Z = X + W $$
The above is the initial statements of a question from chapter 2 of the book Estimation with Applications to Tracking and Navigation.
There is one item that asks for the posterior pdf of the parameter, and here is what I did:
$$ \begin{aligned} p_{X|Z}(x|z) &= \frac{p_{X,Z}(x,z)}{p_Z(z)} \\ &= \frac{p_{Z|X}(z|x)p_X(x)}{p_Z(z)} \\ &= \frac{p_{Z|X}(z|x)p_X(x)}{\int_{-\infty}^{\infty}p_{X,Z}(x,z)dx} \\ &= \frac{p_{Z|X}(z|x)p_X(x)}{\int_{-\infty}^{\infty}p_{Z|X}(z|x)p_X(x)dx}. \end{aligned} $$
PROBABLY THE FOLLOWING IS WRONG. SE EDITED PART!!!
Well $p_X(x)$ is given and $p_{Z|X}(z|x)$ is the convolution $$ \begin{aligned} (p_X(x) * p_W(w))(z) &= \int_{-\infty}^{\infty}p_X(x)p_W(t-x)dx \\ &= \int_{-\infty}^{\infty}p_X(t-w)p_W(w)dw. \end{aligned} $$
With the exposed, I would like to address three questions. Fisrt: Are these steps corrects? Second: If yes, how can I compute the convolution between a continuous and discrete random variables. Third: what would be the posterios pdf of the parameter.
Edited:
I Think that the convolution statement is wrong, because the convolution of the sum between $X$ and $W$ pdfs is $p_Z(z)$ and not $p_Z(z|x)$. So, other question, how can I compute the conditional pdf $P_Z(z|x)$??
Well $p_X(x)$ is given and $p_{Z|X}(z|x)$ is the convolution $$ \begin{aligned} (p_X(x) * p_W(w))(z) &= \int_{-\infty}^{\infty}p_X(x)p_W(t-x)dx \\ &= \int_{-\infty}^{\infty}p_X(t-w)p_W(w)dw. \end{aligned} $$