PROBABILITY SPACES,LIMITS,MEAN VALUE THEOREM,EXPECTATION?
[POSTED QUESTION] a. Determine the limit of? lim s -> t {e^[t X(w]) - e^[s X(w)]} / { t-s} X is a random variable. b. So if Sn is a any sequence of real numbers converging to t, determine the limit lim n-> positive infinity {e^[t X(w]) - e^[Sn X(w)]} / { t-Sn} [BEST ANSWER - Chosen by Asker (ANSWERED BY KB)] a) lim(s→t) {e^[t X(w)] - e^[s X(w)]} / (t - s) = lim(s→t) {e^[s X(w)] - e^[t X(w)]} / (s - t) = (d/ds) e^[s X(w)] {at s = t} = X(w) e^[t X(w)]. b) Since lim(n→∞) sn = t, part (a) yields lim(n→∞) {e^[t X(w)] - e^[sn X(w)]} / (t - sn) = X(w) e^[t X(w)]. [FOLLOW UP-QUESTION] Since w,t and the sequence Sn were arbitrarily chosen, the sequence of real numbers {e^[t X(w]) - e^[Sn X(w)]} / { t-Sn} converges to your answer in (b) for any small omega element of big omega, t element of real numbers and sequence Sn converging to t. c. continuing were we left off in (b),use the Mean Value Theorem on the function f to show that for a fixed positive integer n, there exists a real number On between t and Sn s.t. {e^[t X(w]) - e^[Sn X(w)]} / { t-Sn} = X(w) e ^ (OnX(w)) Again since,small omega,t and Sn and n were arbitrarily chosen,you have shown the existence of a real number On bet Sn and t satisfying the above equation. Assume that the expectations E[e^tX] and E[Xe^tX] are finite for all t element of real numbers and define the function phi: R ->[0,positive infinity) phi(t) = E[e^tX] Show that phi is everywhere differentiable and that for all t element of real nos. phi'(t) = E[Xe^tX] (eqtn 2) d. Show that eqtn 2 may be written lim s->t E[ (e^[tX] - e^[sX] ) / (t-s) ] = E[Xe^tX]
Kang has been diagnosed with attention deficit hyperactivity disorder and recently started a common form of?
drug therapy. Kang is probably receiving a(n) _______ known to _______ the nervous system of adults. barbiturate/speed up barbiturate/slow down stimulant/speed up stimulant/slow down
1st question: Is there design involved in the universE?
1st question: Is there design involved in the universE? as in, the way things work in this universe? the way the different laws compliment each other? 2nd question: if there is design in the universe, shouldn't there be a designer behind the design of the universe? anything that has design will have a designer right?
What's the easy and correct understanding of Random Variable in Statistics?
Although "easy" is certainly relative, I think it is fair to say that what you ask for does not exist. There is an "easy" way to understand random variables that isn't really correct, and there is a correct way to understand them that is not at all easy.The "easy" way that a random variable is explained to students in an introductory level course is that the random variable is the real valued outcome of some random experiment. Simple right? You conduct some experiment whose outcome is not deterministic (i.e. there is more than one possible result of the experiment) and based on whatever the outcome is, your random variable takes on a specific real number. You could randomly sample a student from your class (the experiment) and then write down his or her GPA (on a scale of zero to four). You could throw a fair die and write down the number that appears (an integer from one to six). That's a pretty easy way to understand a random variable right?The problem is, that definition isn't quite enough to allow probability theory to exist in a tidy mathematical framework. Instead, mathematicians define a random variable as a measurable function whose domain is a probability space and whose range is a measurable space. I've underlined three terms that most students don't have the background to understand completely until they are well into a graduate level analysis course (which most of us who have had the pleasure of taking didn't find easy). So if you really want a correct understanding of random variables, the price you pay is that it isn't particularly "easy." On the other hand, if all you want to do is use discrete or continuous probability models to solve some basic problems, then there is no value in mastering the correct understanding. Stick with the easy understanding which is sufficient to allow you to do a lot of interesting work.