Page 1 of 2 12 LastLast
Results 1 to 25 of 39
Like Tree14Likes

Thread: MATH2901 Higher Theory of Statistics

  1. #1
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    MATH2901 Higher Theory of Statistics




  2. #2
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by leehuan View Post


    We can just prove it by cases: either X ≥ Y occurs or Y > X occurs, and in either case the LHS is ≥ 0, since f and g are increasing functions.

    Same is true (replacing X by x and Y by y) if x and y are just real numbers in the domain of f and g if f and g are functions defined on a subset of the reals.
    leehuan likes this.

  3. #3
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: MATH2901 Higher Theory of Statistics

    Whoops. That went over my head.
    Last edited by leehuan; 10 Jun 2017 at 9:22 PM.

  4. #4
    Vexed?
    Join Date
    Sep 2016
    HSC
    N/A
    Gender
    Male
    Location
    Antartica
    Posts
    279
    Rep Power
    2

    Re: MATH2901 Higher Theory of Statistics

    I can't make sense out of the question.

    Is it meant to be (RTP):


  5. #5
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by He-Mann View Post
    I can't make sense out of the question.

    Is it meant to be (RTP):

    No, it was a function of a random variable. But InteGrand's solution worked.
    _______________________________________

    Whilst I get why chisq is the result of Z^2 (Z~N(0,1)), where in practice does it actually get used? What's so powerful about the square of the standard normal?

  6. #6
    Vexed?
    Join Date
    Sep 2016
    HSC
    N/A
    Gender
    Male
    Location
    Antartica
    Posts
    279
    Rep Power
    2

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by leehuan View Post
    No, it was a function of a random variable. But InteGrand's solution worked.
    The point I'm making is that how can a random variable exist alone without specification of it's probability?

    Example, X is gamma random variable with whatever parameters. Prove that X > 0. How does this make sense? It should be prove that P(X > 0) = 1.

  7. #7
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by He-Mann View Post
    The point I'm making is that how can a random variable exist alone without specification of it's probability?

    Example, X is gamma random variable with whatever parameters. Prove that X > 0. How does this make sense? It should be prove that P(X > 0) = 1.
    Look at the 2014 finals. I'm sure you still have it from back when you did the course.

    I don't even care about the distribution of X in this question. I just care that X is a random variable. And I want to find something about f(X), which is what the function does to the random variable (given that f is monotonic increasing).

  8. #8
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by leehuan View Post
    No, it was a function of a random variable. But InteGrand's solution worked.
    _______________________________________

    Whilst I get why chisq is the result of Z^2 (Z~N(0,1)), where in practice does it actually get used? What's so powerful about the square of the standard normal?
    It's used in many hypothesis tests, for example.
    leehuan likes this.

  9. #9
    Vexed?
    Join Date
    Sep 2016
    HSC
    N/A
    Gender
    Male
    Location
    Antartica
    Posts
    279
    Rep Power
    2

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by leehuan View Post
    Look at the 2014 finals. I'm sure you still have it from back when you did the course.

    I don't even care about the distribution of X in this question. I just care that X is a random variable. And I want to find something about f(X), which is what the function does to the random variable (given that f is monotonic increasing).
    Look at another simple example.

    Define f: R -> R such that f(x) = x^2 and let X be a random variable with gamma distribution, say. Prove that f(X) = X^2 > 0.

    It's obvious that this is true because you're just squaring positive random variables and they stay positive. But how does it make sense alone? It should be prove that P(X^2 > 0) = 1.

    The point is, random variables cannot exist alone. It needs to be associated with probabilities.

  10. #10
    Administrator Trebla's Avatar
    Join Date
    Feb 2005
    HSC
    2006
    Gender
    Male
    Posts
    6,140
    Rep Power
    20

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by leehuan View Post
    Whilst I get why chisq is the result of Z^2 (Z~N(0,1)), where in practice does it actually get used? What's so powerful about the square of the standard normal?
    It gets used a lot to understand variances and forms the basis of other distributions such as the F-distribution (which has obvious applications in ANOVA, regression etc)
    leehuan likes this.

  11. #11
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by He-Mann View Post
    Look at another simple example.

    Define f: R -> R such that f(x) = x^2 and let X be a random variable with gamma distribution, say. Prove that f(X) = X^2 > 0.

    It's obvious that this is true because you're just squaring positive random variables and they stay positive. But how does it make sense alone? It should be prove that P(X^2 > 0) = 1.

    The point is, random variables cannot exist alone. It needs to be associated with probabilities.
    It makes perfect sense to me after seeing the earlier question. And again, I couldn't care less if it's normal or gamma or exponential or a discrete r.v.

    You've done the course before. If you're dissatisfied, pick up the exam paper and take it up with the lecturer.

    It might be an abuse of notation. But otherwise I don't see what's wrong with it.

  12. #12
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: MATH2901 Higher Theory of Statistics

    MInd blanking.






  13. #13
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by leehuan View Post
    MInd blanking.





    By the way, this is a Pareto Distribution with scale parameter 1 (and shape parameter alpha). This distribution is related to the "80-20" law or "Pareto principle", which you may have heard of.

    A fact about Pareto distributions is that the conditional distribution of a Pareto r.v. X given the event that X is greater than a given number b (where b is greater than or equal to the scale parameter of the distribution) is still a Pareto distribution, with the same shape parameter but with new scale parameter b. Using this fact and the formula for the mean of a Pareto distribution, the desired conditional mean can be deduced.

    (These facts are facts that should be proved before being used for this Q. I suppose, and can be all proven as an exercise using standard methods for dealing with truncated distributions. For this particular Q., you wouldn't need to know these facts, you could just do the Q. normally, but they are interesting and it is what the Q. was probably getting at (particularly the 80-20 law, which is maybe why they chose the 80-th percentile), so I included them here.)
    Last edited by InteGrand; 18 Jun 2017 at 6:41 AM.

  14. #14
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: MATH2901 Higher Theory of Statistics








  15. #15
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by leehuan View Post






    leehuan likes this.

  16. #16
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by InteGrand View Post
    For my reference sake, does the 2,2 entry of the inverse approximate Var(\hat{\beta})?


    Edit: Oops.
    Last edited by leehuan; 18 Jun 2017 at 3:22 PM.

  17. #17
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by leehuan View Post
    For my reference sake, does the 2,2 entry of the inverse approximate \hat{\beta}?
    leehuan likes this.

  18. #18
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: MATH2901 Higher Theory of Statistics

    Hang on, I'm not sure if I'm doing something wrong because I run into a circular argument.



    Last edited by leehuan; 18 Jun 2017 at 3:32 PM.

  19. #19
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by leehuan View Post
    Hang on, I'm not sure if I'm doing something wrong because I run into a circular argument.



    I think double check your Var, the beta terms should end up cancelling out I think.
    leehuan likes this.

  20. #20
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by InteGrand View Post
    I think double check your Var, the beta terms should end up cancelling out I think.
    Oh. My bad. I didn't do 1/det

  21. #21
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: MATH2901 Higher Theory of Statistics

    I think you forgot to divide by the determinant of the Fisher Information Matrix. You appear to have multiplied by it (or something).
    leehuan likes this.

  22. #22
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: MATH2901 Higher Theory of Statistics

    Also I think the final answer's denominator should be n^2 (not n).

  23. #23
    Supreme Member Flop21's Avatar
    Join Date
    May 2013
    HSC
    2015
    Gender
    Female
    Posts
    2,850
    Rep Power
    5

    Re: MATH2901 Higher Theory of Statistics

    Any tips on what test statistic to use for a hypothesis test?
    2015 HSC: English Adv, Mathematics, Business Studies, Biology, Multimedia.

    HSC Biology Flashcards

  24. #24
    Rambling Spirit
    Join Date
    Dec 2014
    HSC
    N/A
    Gender
    Male
    Posts
    6,039
    Rep Power
    7

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by Flop21 View Post
    Any tips on what test statistic to use for a hypothesis test?
    It depends on the hypothesis test and the situation at hand. Common ones are when you have asymptotic normality under a null hypothesis (which is often the case from the Central Limit Theorem), you can use a Z-test ( https://en.wikipedia.org/wiki/Z-test ), or you could use a Chi-Squared test sometimes. Depends on what purpose the test is serving.

  25. #25
    Ancient Orator leehuan's Avatar
    Join Date
    May 2014
    HSC
    2015
    Gender
    Male
    Posts
    5,807
    Rep Power
    6

    Re: MATH2901 Higher Theory of Statistics

    Quote Originally Posted by InteGrand View Post
    Also I think the final answer's denominator should be n^2 (not n).
    Oh I double checked that. I put the n back in there, got a determinant of (2alpha beta^2)/n^2 * (a matrix with n's appearing everywhere). That's fine I reckon

    Quote Originally Posted by Flop21 View Post
    Any tips on what test statistic to use for a hypothesis test?
    When we were taught it, we were honestly taught "educated guess". Normally I try to exploit the CLT.
    Flop21 likes this.

Page 1 of 2 12 LastLast

Thread Information

Users Browsing this Thread

There are currently 1 users browsing this thread. (0 members and 1 guests)

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •