Page 1 of 2 12 LastLast
Results 1 to 15 of 27
Like Tree4Thanks

Math Help - Normal distribution and unit

  1. #1
    Junior Member
    Joined
    Mar 2010
    Posts
    32

    Normal distribution and unit

    I think the unit of PDF of normal distribution should be unitless or dimensionless.

    http://upload.wikimedia.org/math/d/6...9d037a126c.png

    But, I found a sigma in the denominator of PDF of normal distribution. So, the sigma should be unitless.

    But, the sigma is the standard deviation of the random variable. This means the sigma and the random variable are also unitless.

    Is this correct?

    If so, can we apply normal distribution to any random variable with unit, e.g. meter or kg?

    If we can do this, the unit of PDF of normal distribution would be 1/meter or 1/kg due to the existence of sigma in the equation.

    I think this is not a correct expression of unit of probability.

    Could any one help explain?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    3,619
    Thanks
    592

    Re: Normal distribution and unit

    Hey eulerian.

    What you are dealing with functions of any kind, things are chosen to be dimension-less if you are dealing with transcendental functions (like exponentials, trig functions, and others).

    The distribution itself is going to represent a probability which is always unit-less (it is a number). If you are talking about some probability with respect to another unit, then that is a different quantity in itself.

    For more information, you should probably read an applied math book, engineering book, or physics book on units and they come into play with functions, transformations, and relations to other units.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,344
    Thanks
    899

    Re: Normal distribution and unit

    Quote Originally Posted by eulerian View Post
    I think the unit of PDF of normal distribution should be unitless or dimensionless.

    http://upload.wikimedia.org/math/d/6...9d037a126c.png

    But, I found a sigma in the denominator of PDF of normal distribution. So, the sigma should be unitless.

    But, the sigma is the standard deviation of the random variable. This means the sigma and the random variable are also unitless.

    Is this correct?

    If so, can we apply normal distribution to any random variable with unit, e.g. meter or kg?

    If we can do this, the unit of PDF of normal distribution would be 1/meter or 1/kg due to the existence of sigma in the equation.

    I think this is not a correct expression of unit of probability.

    Could any one help explain?
    let p(x) be any probability distribution. p(x) dx represents an infinitesimal probability and thus is unitless. But dx represents a infinitesimal bit of a random variable that can have units. Thus p(x) must have units 1/(units of x).

    For the Normal distribution you can see this from the standard deviation in the denominator. The std has units of the underlying random variable. The exponential term is unitless. So we end up with 1/(units of std) = 1/(units of underlying rv) as the units of p(x).
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Mar 2010
    Posts
    32

    Re: Normal distribution and unit

    Quote Originally Posted by romsek View Post
    let p(x) be any probability distribution. p(x) dx represents an infinitesimal probability and thus is unitless. But dx represents a infinitesimal bit of a random variable that can have units. Thus p(x) must have units 1/(units of x).

    For the Normal distribution you can see this from the standard deviation in the denominator. The std has units of the underlying random variable. The exponential term is unitless. So we end up with 1/(units of std) = 1/(units of underlying rv) as the units of p(x).

    Thank you.

    So, the unit of likelihood function should also be 1/(units of underlying rv).

    As I know, log function is a transcendental function, which only deals with unitless variables.

    What is the unit of log-likelihood function? Can we log 1/(units of underlying rv)?
    Last edited by eulerian; November 30th 2013 at 11:26 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Mar 2010
    Posts
    32

    Re: Normal distribution and unit

    Quote Originally Posted by chiro View Post
    Hey eulerian.

    What you are dealing with functions of any kind, things are chosen to be dimension-less if you are dealing with transcendental functions (like exponentials, trig functions, and others).

    The distribution itself is going to represent a probability which is always unit-less (it is a number). If you are talking about some probability with respect to another unit, then that is a different quantity in itself.

    For more information, you should probably read an applied math book, engineering book, or physics book on units and they come into play with functions, transformations, and relations to other units.
    Do you agree with #3 and #4?

    Thank you
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,344
    Thanks
    899

    Re: Normal distribution and unit

    Quote Originally Posted by eulerian View Post
    Thank you.

    So, the unit of likelihood function should be 1/(units of underlying rv).

    As I know, log function is a transcendental function, which only deals with unitless variables.

    What is the unit of log-likelihood function? Can we log 1/(units of underlying rv)?
    If I recall the likelihood function is the ratio of two of the same type of distribution and thus will be unitless as it must be to enable the log likelihood function to exist.

    The log likelihood function will be unitless as well.
    Thanks from eulerian
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Mar 2010
    Posts
    32

    Re: Normal distribution and unit

    Quote Originally Posted by romsek View Post
    If I recall the likelihood function is the ratio of two of the same type of distribution and thus will be unitless as it must be to enable the log likelihood function to exist.

    The log likelihood function will be unitless as well.

    Thank you so much.

    So, in maximum-likelihood estimation, the sigma must be unitless and be a ratio?

    http://upload.wikimedia.org/math/4/8...8f7eb26b76.png
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,344
    Thanks
    899

    Re: Normal distribution and unit

    Quote Originally Posted by eulerian View Post
    Thank you so much.

    So, in maximum-likelihood estimation, the sigma must be unitless and be a ratio?

    http://upload.wikimedia.org/math/4/8...8f7eb26b76.png
    it's been a long time since I looked at something like that. I'm going to have to review it a bit.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Mar 2010
    Posts
    32

    Re: Normal distribution and unit

    Any opinion is appreciated
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,344
    Thanks
    899

    Re: Normal distribution and unit

    It looks like they are just taking the log of the standard deviation w/o regard to it's units. I'd have to read up on this a bit to give you a justification. This formula is for finding the maximum likelihood estimates of the mean and variance given a set of sample data?

    I don't think I'd worry too much about the units of the likelihood function. It's not a physical thing.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Junior Member
    Joined
    Mar 2010
    Posts
    32

    Re: Normal distribution and unit

    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,344
    Thanks
    899

    Re: Normal distribution and unit

    Quote Originally Posted by eulerian View Post
    Ok, I see. The log of the distribution is basically the entropy of the underlying random variable. I guess it's units would be in nats.

    I'm not seeing any discussions of the units of the log-likelihood function jumping out at me off the web.
    Last edited by romsek; December 1st 2013 at 01:00 AM.
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Junior Member
    Joined
    Mar 2010
    Posts
    32

    Re: Normal distribution and unit

    Also, if 1/(units of underlying rv) is the units of p(x), how can we get the below?

    http://upload.wikimedia.org/math/9/f...4dfaedc20e.png

    Because Σ(p(x)x)=E[X], the unit should be $, not unitless.

    Thank you.
    Follow Math Help Forum on Facebook and Google+

  14. #14
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,344
    Thanks
    899

    Re: Normal distribution and unit

    Quote Originally Posted by eulerian View Post
    Also, if 1/(units of underlying rv) is the units of p(x), how can we get the below?

    http://upload.wikimedia.org/math/9/f...4dfaedc20e.png

    Because Σ(p(x)x)=E[X], the unit should be $, not unitless.

    Thank you.
    For continuous rvs you have E[X] = Integral[x p(x) dx, -inf, inf] and that dx contributes units.

    for discrete rvs the pk's are probabilities, not probability densities and thus are unitless. Thus Sum[k pk, 0, inf] has units of k, in this case dollars
    Thanks from eulerian
    Follow Math Help Forum on Facebook and Google+

  15. #15
    Junior Member
    Joined
    Mar 2010
    Posts
    32

    Re: Normal distribution and unit

    Quote Originally Posted by romsek View Post
    For continuous rvs you have E[X] = Integral[x p(x) dx, -inf, inf] and that dx contributes units.

    for discrete rvs the pk's are probabilities, not probability densities and thus are unitless. Thus Sum[k pk, 0, inf] has units of k, in this case dollars
    Thank you so much again.

    I get it.


    But, I have not learned information entropy. Could you just show out some dimensional analysis about the operation between any other units and the units of information(e.g. bits or nats)?
    Follow Math Help Forum on Facebook and Google+

Page 1 of 2 12 LastLast

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: February 15th 2013, 01:01 PM
  2. Calculating unit normal
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: January 9th 2013, 11:41 AM
  3. Replies: 3
    Last Post: October 22nd 2012, 08:57 AM
  4. Unit normal
    Posted in the Geometry Forum
    Replies: 4
    Last Post: September 10th 2010, 08:23 PM
  5. Unit Normal
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 18th 2008, 05:08 AM

Search Tags


/mathhelpforum @mathhelpforum