Just a quick question on the definition of the exponential distr. function. Is it a mistake to define the function seperately for x<0, x=0 and x>0? Why is it necessary to define only for 2 conditions of x< and x>=0? Is it because x is R?
Also if anybody could explain to me why taking a sup and not calculating derivatives of the functions....
Once you cover the full range, it is valid. 3 groups as you outlined and corresponding formulae (once correct) is OK, but it's more efficient to use 2, ie x<0 and x>=0; Clearly you need at least 2 groups since you're finding the inverse of something with |x| Not sure what your second question relates to.
Thank you that clarifies my concerns.... My second question relates to finding M as the maximum of the ratio of the two functions. In the core reading and examples, I would typically take a derivative of both functions..... Why is the solution not taking derivatives of function f and function g but taking a "supremum" = sup of the ratio?
Essentially you need to make sure the top function is always above the bottom one. Finding individual maximums isn't too helpful since they could occur at different points and the bottom function may exceed the top at some other point than the max. So you have to consider the ratio as a whole
so the idea is to simplify the ratio of two functions and then take a derivative of this ratio which should be consistent with the supremium?
You need to make the ratio <=1. Dividing by the maximum value of the ratio would do so, across the curve. This is essentially the supremum value. Obviously dividing by a larger number would have the same effect but would be less efficient. Eg if you divide by 10^100, it would be valid but would take you forever to find a simulation that is accepted. The supremum value is simply the most efficient number, ie smallest, which satifies the ratio <= 1. From there, differentiation is one (but probably most common) of the many techniques available to find such a value.
