Problem is here 2(c)
Solution is here 2(c)
$$\quad \ Z\ =\ Y\ -\ X\quad\quad; 0<Y<2x$$ $$\quad\quad\ => Z\ =\ Y\ -\ X\quad\quad; -x<Y\ -\ X<x$$ $$=> Z\ =\ Y\ -\ X\quad\quad; -x<Z<x$$ $$ f_{X,Z}(x,z)= f_X(x)f_{Z|X}(z|x)\quad; -x<Z<x$$ We are given that Y is uniformly distributed, if we fix X then Z will also have the same distribution as Y but shifted by a constant.
Y has a Uniform distribution of $1/2x$.
$$f_{Y|X}(y |x)= \frac{1}{2x}\quad; 0<Y<2x$$ So Z should also have the same distridution but shifted by a constant say 'c'
$$f_{Z|X}(z |x)= \frac{1}{2x} - c\quad; -x<Z<x$$

But the solution says: $$f_{Z|X}(z |x)= \frac{1}{2x}\quad; -x<Z<x$$ Whats am missing?


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