Hint for part a. Do you know the mean value theorem for integrals?
let f be a continuous density which is zero outside in the interval[0,A]
,where A >0, let F be the corresponding cdf.
a) prove that there exists at least one point X such that f(x)=1/A
b), let 0<p<1 prove there exists at least one point Xp such that Xp is the pth quantile of the distribution defined by f and F, give an example to show that the point Xp may not be unique.
Thanks a lot!!
You can't prove it- it's not true. For example, if A= 1/2, F(x) is always between 0 and 2 while 1/A= 2.
What do you mean by "quantile"? A point where the probability P(x< Xp)= p? In that case you just need the fact that F is a continuous function, F(0)= 0, F(A)= 1 and the "intermediate value property" of continuous functions.b), let 0<p<1 prove there exists at least one point Xp such that Xp is the pth quantile of the distribution defined by f and F, give an example to show that the point Xp may not be unique.
Thanks a lot!!