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Thread: Normal distributions?

  1. #1 Normal distributions? 
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    Given a normal distribution of the weight of people that wake up early:



    Where is the variance and is the mean.

    And a normal distribution of the weight of people that go to sleep late:



    Where is the variance and is the mean.

    Is there any way to find the normal distribution of people that wake up early AND go to sleep late?

    My wild guess would be


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  3. #2  
    Forum Freshman Fatboy Miller's Avatar
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    If anything it would be:
    Fx . Gx
    which would be the independent distribution.

    There is overlap between these two (as shown by a Venn diagram - one circle represents Fx, the other Gx, where they overlap is fx and gx.



    It depends on the dependence in question!!! And so there is no specific formula other than is 20% of people who wake up early also go to bed late then the distibution can be written as:
    0.2 . Fx
    In reliability engineering the '0.2' (or whatever the dependence is) is called the beta factor. Not sure if there is the same / equivalent in statistics.

    Hope this helps!


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    Quote Originally Posted by Fatboy Miller
    It depends on the dependence in question!!! And so there is no specific formula other than is 20% of people who wake up early also go to bed late then the distibution can be written as:
    0.2 . Fx
    In reliability engineering the '0.2' (or whatever the dependence is) is called the beta factor. Not sure if there is the same / equivalent in statistics.

    Hope this helps!
    I'm a little bit confused by this! So you're saying that, if I were to tell you that the dependence was 20%, then the resulting distribution wouldn't incorporate both of the functions? Only f(x)? This is a bit counter-intuitive to me, I think I've missed something.

    Or do you mean the resulting distribution would be 0.2f(x)g(x)?
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  5. #4 Re: Normal distributions? 
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    Quote Originally Posted by m84uily
    Is there any way to find the normal distribution of people that wake up early AND go to sleep late?
    You cannot say with certainty that any such people exist. If they do, you do not know that their weight is normally distributed.

    Going to sleep late and waking up early may not be, in fact probably are not, independent.
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  6. #5 Re: Normal distributions? 
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by m84uily
    Is there any way to find the normal distribution of people that wake up early AND go to sleep late?
    You cannot say with certainty that any such people exist. If they do, you do not know that their weight is normally distributed.

    Going to sleep late and waking up early may not be, in fact probably are not, independent.
    Okay, thanks!
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    Forum Freshman Fatboy Miller's Avatar
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    Let me try to re-phrase my waffle. If getting up early had no relation to what time you went to bed (i.e. independence) then it is:
    Fx.Gx.

    However, by intuition, those who got up early are MORE unlikely to be those who went to bed late (i.e. dependent). The only way of doing it that I know of... ...is to have a probability of 'P' (i.e. going to bed late) given distribution Fx (i.e. got up early). Therefore it is distribution Fx 'lowered' by a factor of 'P'.

    P . Fx

    Where in my example P = 0.2 (i.e. probability of going to bed late GIVEN that they got up early).

    Hope this clarifies my point.
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  8. #7  
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    Quote Originally Posted by Fatboy Miller
    Let me try to re-phrase my waffle. If getting up early had no relation to what time you went to bed (i.e. independence) then it is:
    Fx.Gx.

    However, by intuition, those who got up early are MORE unlikely to be those who went to bed late (i.e. dependent). The only way of doing it that I know of... ...is to have a probability of 'P' (i.e. going to bed late) given distribution Fx (i.e. got up early). Therefore it is distribution Fx 'lowered' by a factor of 'P'.

    P . Fx

    Where in my example P = 0.2 (i.e. probability of going to bed late GIVEN that they got up early).

    Hope this clarifies my point.
    Yes that helps a lot, thanks!
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  9. #8  
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    Quote Originally Posted by Fatboy Miller
    Let me try to re-phrase my waffle. If getting up early had no relation to what time you went to bed (i.e. independence) then it is:
    Fx.Gx.

    However, by intuition, those who got up early are MORE unlikely to be those who went to bed late (i.e. dependent). The only way of doing it that I know of... ...is to have a probability of 'P' (i.e. going to bed late) given distribution Fx (i.e. got up early). Therefore it is distribution Fx 'lowered' by a factor of 'P'.

    P . Fx

    Where in my example P = 0.2 (i.e. probability of going to bed late GIVEN that they got up early).

    Hope this clarifies my point.
    There is a wee problem with all of your suggested densities -- none of them integrate to 1.
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    Forum Freshman Fatboy Miller's Avatar
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    Well spotted...

    ..but it integrates to P (0.2 in the hypothesised example). So when integrated you do get the probability that some went to bed late and got up early.
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  11. #10  
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    Quote Originally Posted by Fatboy Miller
    Well spotted...

    ..but it integrates to P (0.2 in the hypothesised example). So when integrated you do get the probability that some went to bed late and got up early.
    That was not the question,

    The OP asked for the probability density function of the weight of people who both went to bed late and got up early.
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  12. #11  
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    Now I understand where all my confusion was coming from XD
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  13. #12  
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    But you could can't come up with a definitive distribution as the "X-axis" would be measuring two distinct parameters (i.e. if I went to bed at 11 O'clock and got up at 7 O'clock my data would be in two places at once).

    I conside that my approach is using a bit of 'fudge' and is 'arse about tit' as you effectively need the answer to the original question in order to answer it (deriving a probability that those who got up early also went to bed late means that you're effectively at the end point anyway!).

    I also concede that my solution does not give a true probability density as it discounts those who did not go to bed late.

    All I did was to attempt to outline an alternative approach (i.e. having the original density of Fx and then applying a probability which is related to Gx and it's dependence with Fx). It's the only way I can see of coming up with a reasonable answer...
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  14. #13  
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    Quote Originally Posted by Fatboy Miller
    But you could can't come up with a definitive distribution as the "X-axis" would be measuring two distinct parameters (i.e. if I went to bed at 11 O'clock and got up at 7 O'clock my data would be in two places at once).

    I conside that my approach is using a bit of 'fudge' and is 'arse about tit' as you effectively need the answer to the original question in order to answer it (deriving a probability that those who got up early also went to bed late means that you're effectively at the end point anyway!).

    I also concede that my solution does not give a true probability density as it discounts those who did not go to bed late.

    All I did was to attempt to outline an alternative approach (i.e. having the original density of Fx and then applying a probability which is related to Gx and it's dependence with Fx). It's the only way I can see of coming up with a reasonable answer...
    The x-axis would be weight, the y axis would describe the percent of people that weigh that much.
    What about, if we assume h(x) describes people who got up early and went to bed late (they exist) and is a normal distribution that has variance and mean
    Would that be correct?
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  15. #14  
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    Quote Originally Posted by m84uily
    Quote Originally Posted by Fatboy Miller
    But you could can't come up with a definitive distribution as the "X-axis" would be measuring two distinct parameters (i.e. if I went to bed at 11 O'clock and got up at 7 O'clock my data would be in two places at once).

    I conside that my approach is using a bit of 'fudge' and is 'arse about tit' as you effectively need the answer to the original question in order to answer it (deriving a probability that those who got up early also went to bed late means that you're effectively at the end point anyway!).

    I also concede that my solution does not give a true probability density as it discounts those who did not go to bed late.

    All I did was to attempt to outline an alternative approach (i.e. having the original density of Fx and then applying a probability which is related to Gx and it's dependence with Fx). It's the only way I can see of coming up with a reasonable answer...
    The x-axis would be weight, the y axis would describe the percent of people that weigh that much.
    What about, if we assume h(x) describes people who got up early and went to bed late (they exist) and is a normal distribution that has variance and mean
    Would that be correct?

    You are essentially stuck unless you can assume that The sigma-algebras generated by measurable subsets of "go to bed late" and "wake up early" are independent, in which case the densities are the original densities.

    Given no known relationship between the two classes there is nothing that can be said. It is quite possible, for instance that they have empty intersetion. If there were only one person in the intersection your density would be a delta function.
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