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We have given parameter family over over $\{0,1\}\times \mathbb{R}$ where random variables $X,Y$ are such that $P(Y=0)=\frac{1}{2}=P(Y=1)$ and we know that $X|Y=0$ is Normal with unknown mean $\theta$ and variance equal $1$ and $X|Y=1$ is Normal with mean $\theta$ and variance $2$. Calculate likelihood function. Thanks in advance

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    There is no joint PDF since Y is discrete. – 2017-01-03
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    how about the mixed case like here https://en.wikipedia.org/wiki/Joint_probability_distribution#Mixed_case – 2017-01-03
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    Indeed, what about it? Seems like just another misleading WP page (note the strange use of quotes around the word mixed). Anyway, *what is your question exactly?* – 2017-01-03
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    I have parametric family for $\theta \in \mathbb{R}$ over $\{0,1\}\times \mathbb{R}$ and I need to find likelihood function for $\theta$ – 2017-01-03
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    Then ask *this* and not the non-question above. – 2017-01-03

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you can calculate the joint CDF as follows $$ F_{X,Y}(x,y) = \mathbb P(X\leq x, Y\leq y)= \begin{cases} 0, & y<0, \\[6pt] \frac 1 {\sqrt{2\pi}} \exp \bigl(-\frac 12 (x-\theta)^2\bigr), & y \in [0,1),\\[6pt] F_X(x), & y\geq 1, \end{cases} $$ where $F_X$ is the marginal PDF of $X$: \begin{align} F_X(x) &= \mathbb P (X \leq x) = \mathbb P( X \leq x\mid Y=0)\cdot \mathbb P(Y=0) X+\mathbb P( X \leq x\mid Y=1)\cdot \mathbb P(Y=1) \\ &=\frac 12 \Phi(x-\theta) + \frac 12 \Phi\biggl(\frac{x-\theta}{\sqrt{2}}\biggr), \end{align} where $\Phi$ is the standard Gaussian CDF.

The joint pdf does not exist, since $Y$ is discrete and only continuous distributions have a density.

The liklihood function is $$ L(\theta;x_1,\dots,x_n)= \frac 1{2^n} \prod_{i=1}^n \Bigl( \frac 1 {\sqrt{2\pi}} \exp \bigl(-\frac 12 (x_i-\theta)^2\bigr)+ \frac 1 {\sqrt{4\pi}} \exp \bigl(-\frac 14 (x_i-\theta)^2\bigr) \Bigr) $$

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    The only interesting piece in this would be (perhaps) the one you skip, namely, the value of $F_X(x)$... – 2017-01-03
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    only because in the question it says that "I know how to calculate marginal PDF for X" – 2017-01-03