What do i have to use to calculate the probablity that the range of a random sample of size $3$ from the uniform distribution is less than $0.8$?
I have the pdf of the range :
$$f_W(w)= n(n-1) \int_\infty^\infty [F(x+w)-F(x)]^{n-2} f(x) f(x+w)dx $$
and i must calculate $$ P(U_s-U_r<0.80)$$
Is that right?
You wish to calculate the probability that the maximum of the three values is less than 0.8 more than their minimum; When the three values are samples drawn iid from a uniform $(0;1)$ distribution this is:
$$\mathsf P(X_{(3)}-X_{(1)}<0.8) \;=\; 3\,\int_0^1\left(\int_{\max(0, x-0.8)}^x\operatorname d y\right)^2 \operatorname d x \\ = \;3\left(\int_0^{0.8}(x)^2\operatorname d x+\int_{0.8}^1(0.8)^2\operatorname d x\right)$$