i am doing my statistical inference class. I want to know that the uniform distribution on [a,b] belongs to location and scale family.

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#### Best Answer

For any given distribution (on $mathbb{R}$, say, or an interval) $f_0$, we can *generate* a location-scale family by $$ f(x ; mu,sigma) =frac1{sigma} f(frac{x-mu}{sigma}) $$ The distribution family most often considered with the uniform distribution is $U(0, theta), theta > 0$ which is a one-parameter family and so cannot be written in location-scale form. If you with your question consider the family $U(a,b),a<b$ where $a,b$ otherwise are free, then you can write that in scale-location form. You can for example chose for $f_0$ $$ f_0(x)= frac12, quad -1 le x le 1 $$

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