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Thread: Statistics Marathon & Questions

  1. #151
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    Re: University Statistics Discussion Marathon

    Just want to confirm if this is the right answer :

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    Re: University Statistics Discussion Marathon

    Quote Originally Posted by davidgoes4wce View Post
    Just want to confirm if this is the right answer :

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  3. #153
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    Re: Statistics

    How do you find the marginal distribution of a random variable, given a table of the joint probabilities of that random variable and another?

    i.e. fY(y).

    I see the solution but not sure where they are getting it from.
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    Re: Statistics

    Quote Originally Posted by Flop21 View Post
    How do you find the marginal distribution of a random variable, given a table of the joint probabilities of that random variable and another?

    i.e. fY(y).

    I see the solution but not sure where they are getting it from.
    You integrate or sum the joint density function over all values of the other variable(s).

  5. #155
    Ancient Orator leehuan's Avatar
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    Re: Statistics

    Quote Originally Posted by Flop21 View Post
    How do you find the marginal distribution of a random variable, given a table of the joint probabilities of that random variable and another?

    i.e. fY(y).

    I see the solution but not sure where they are getting it from.








    Which is nothing different to what IG said. He basically just described a general case where you have n r.v.s

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    Ancient Orator leehuan's Avatar
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    Re: Statistics



    And we're using the convention X ~ exp(a) => fX(x) = 1/a exp(-x/a)





    But how do you find the conditions on y1 and y2 where this density is defined for?

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    Re: University Statistics Discussion Marathon

    50 % of the meals sold at Graylands Hospital contain rice.

    Answer the following question, keeping in mind that your answer must be a number between 0 and 1, i.e do not use percentages. Round off the final result to 2dp, but don't round numbers during your calculations.

    What is the probability that out of 78 meals sold at Graylands Hospital, 43 or more contain rice?
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    Re: Statistics

    Quote Originally Posted by leehuan View Post


    And we're using the convention X ~ exp(a) => fX(x) = 1/a exp(-x/a)





    But how do you find the conditions on y1 and y2 where this density is defined for?
    Last edited by InteGrand; 21 Apr 2017 at 9:21 AM. Reason: Typo
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    Re: Statistics

    Last edited by InteGrand; 20 Apr 2017 at 8:57 PM. Reason: Typo
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    Re: University Statistics Discussion Marathon

    Quote Originally Posted by davidgoes4wce View Post
    50 % of the meals sold at Graylands Hospital contain rice.

    Answer the following question, keeping in mind that your answer must be a number between 0 and 1, i.e do not use percentages. Round off the final result to 2dp, but don't round numbers during your calculations.

    What is the probability that out of 78 meals sold at Graylands Hospital, 43 or more contain rice?

    Not sure... how to compute... (p = 0.5)

    Last edited by He-Mann; 21 Apr 2017 at 12:36 AM.

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    Re: University Statistics Discussion Marathon

    Quote Originally Posted by He-Mann View Post
    Not sure... how to compute... (p = 0.5)

    what value of p did you get?
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    Re: University Statistics Discussion Marathon

    Quote Originally Posted by davidgoes4wce View Post
    what value of p did you get?
    Do you mean the value of the sum? (The "p" He-Mann used was referring to the success probability parameter, which is 0.5.)
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    Ancient Orator leehuan's Avatar
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    Re: Statistics

    U1, ..., Un are iid Unif(0,1)




    So I followed their working up to






    ______________________

    Edit: Actually my bad on transcription error, their working stopped here

    Last edited by leehuan; 21 Apr 2017 at 5:49 PM.

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    Re: Statistics

    Quote Originally Posted by leehuan View Post
    U1, ..., Un are iid Unif(0,1)




    So I followed their working up to






    ______________________

    Edit: Actually my bad on transcription error, their working stopped here

    The integral you were thinking about is actually the "complementary CDF" or "survival function" (though the limits aren't right and I realised I misinterpreted your query before seeing your addendum), which is the probability Y_n > y. It's actually neater here find this survival function first and then just get the CDF of Y_n from that by taking its complement (subtracting it from 1). The reason it's nicer to work with the survival function here is that's we're dealing with a min function, and the minimum of some numbers is greater than something iff all the numbers are greater than that thing. If instead we were working with a max function, it'd be nicer to use the CDF, because a maximum of some numbers is less than or equal to something iff all the numbers are less than or equal to that thing.

    Edit: I think I misinterpreted your query (I think I get it now after seeing your addendum). I addressed what I think was your query in my next post.
    Last edited by InteGrand; 21 Apr 2017 at 6:06 PM.
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    Re: Statistics

    Quote Originally Posted by leehuan View Post
    U1, ..., Un are iid Unif(0,1)




    So I followed their working up to






    ______________________

    Edit: Actually my bad on transcription error, their working stopped here

    Last edited by InteGrand; 22 Apr 2017 at 10:44 AM.
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  16. #166
    Ancient Orator leehuan's Avatar
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    Re: Statistics

    Quote Originally Posted by InteGrand View Post
    Got it, thanks, 90% sure that was my confusion

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    Ancient Orator leehuan's Avatar
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    Re: Statistics

    I computed an MGF to be



    Therefore it does not exist because it's not defined for any u in [interval containing 0] right?

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    Re: Statistics

    Quote Originally Posted by leehuan View Post
    I computed an MGF to be



    Therefore it does not exist because it's not defined for any u in [interval containing 0] right?
    Actually that function has finite limit as u -> 0, as you can show using L'Hôpital's rule for example. So the MGF does exist.
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    Ancient Orator leehuan's Avatar
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    Re: Statistics

    Quote Originally Posted by InteGrand View Post
    Actually that function has finite limit as u -> 0, as you can show using L'Hôpital's rule for example. So the MGF does exist.
    Nice. I knew I was missing something.

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    Re: University Statistics Discussion Marathon

    Can't any quick way of doing it at the moment. Please let me know if any of this is wrong.



    Since probabilities sum to 1, we have



    By symmetry (i.e. Pascal's Triangle), we have



    Then from (*), we have



    Now, we compute the desired probability,



    In retrospect, I should have wrote p = 0.5 = 1 - p and combined the product which would make it easier to read.
    Last edited by He-Mann; 28 Apr 2017 at 9:13 PM.

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    Ancient Orator leehuan's Avatar
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    Re: Statistics







    Once again stuck on domains. I keep forgetting how linear transforms map regions so maybe just another 'by inspection' method please?

    (Intuitive guess: -60 < z < 60. Just don't know how to get there)

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    Re: Statistics

    Quote Originally Posted by leehuan View Post






    Once again stuck on domains. I keep forgetting how linear transforms map regions so maybe just another 'by inspection' method please?

    (Intuitive guess: -60 < z < 60. Just don't know how to get there)
    It's quite easy with the linear transforms method. The sample space in the v-w space is just a square. A square gets mapped to a parallelogram in the u-z space (in general) by a linear transformation like the one you have. (Recall that in general, an invertible linear map from Rn to Rn will map lines to lines and parallelograms to parallelograms.)

    To find the coordinates of the vertices this parallelogram, just find what points each of the vertices of the square in the v-w sample space get mapped to by the linear transformation. Then the interior of the sample space in the v-w space gets mapped to the inside of the parallelogram with those vertices in the u-z space.
    Last edited by InteGrand; 22 Apr 2017 at 5:22 PM.
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    Re: Statistics

    Quote Originally Posted by leehuan View Post






    Once again stuck on domains. I keep forgetting how linear transforms map regions so maybe just another 'by inspection' method please?

    (Intuitive guess: -60 < z < 60. Just don't know how to get there)
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  24. #174
    Ancient Orator leehuan's Avatar
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    Re: Statistics

    That's interesting, u - 60 < z \le u was the first thing I had.


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    Re: Statistics

    Quote Originally Posted by leehuan View Post
    That's interesting, u - 60 < z \le u was the first thing I had.

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