Let $X$ and $Y$ be jointly distributed random variables such that the conditional distribution of $Y$ given $X =x $ is uniform in the interval $(x-1,x+1)$.
Now we are given $E(X) = 1$ and $Var(x) = \frac{5}{3}$, now how can we proceed to calculate expectation of random variable $Y$ and variance of $Y$ ?
I can only start $f(y|X= x) = \frac{f_{X,Y}(x,y)}{f_{X}(x)}$
as we have a continuous uniform distribution now here perhaps I should write $x-1<y<x+1$.
I am stuck here,how to extract information from $Y$ ?
Given $Y\mid X \sim\mathcal U(X-1;X+1)$ you should know, or be able to find, the following: $${\mathsf E(Y\mid X)\\ \mathsf{Var}(Y\mid X)}$$
Also given $\mathsf E(X)=1$ and $\mathsf {Var}(X)=5/3$, you can use these four evaluations with the following:
Law of Total Expectation: $$\mathsf E(Y) = \mathsf E(\mathsf E(Y\mid X))$$
Law of Total Variance: $$\mathsf {Var}(Y) = \mathsf E(\mathsf {Var}(Y\mid X))+\mathsf {Var}(\mathsf E(Y\mid X))$$