Gundala Viswanath
2009-Sep-07 06:42 UTC
[R] How Does One Use the Value of Density Function?
How do people usually use the result of density function (e.g. dnorm)? Especially when its value can be greater than 1. What do they do with such density >1?> dnorm(2.02,2,.24)[1] 1.656498 - G.V.
On 07-Sep-09 06:42:06, Gundala Viswanath wrote:> How do people usually use the result of density function (e.g. dnorm)? > Especially when its value can be greater than 1. > > What do they do with such density >1? > >> dnorm(2.02,2,.24) > [1] 1.656498 > > - G.V.The point is that it is a *density* of probability. The greater the density, the more compactly a given amount of probability is distributed over a given range of X (or, the narrower the range of X ove which a given amount of probability is distributed); the lower the density, the more widely it is dispersed. The concept of density is, in effect, the amount of probability per unit length in an interval, so "division by length" is part of it. To convert probability density to probability, multiply by a length. To obtain (a close approximation to) the amount of probability within a short range, such as 2.01 to 2.03, when X has your Normal distribution with mean 2.02 and SD 0.24, multiply the value of the density function at the midpoint 2.02 by the length 0.02 of the interval: dnorm(2.02,2,.24)*0.02 # = 0.03312996 and compare it with the amount of probability calculated from pnorm: pnorm(2.03,2,.24) - pnorm(2.01,2,.24) # = 0.03312044 The approximation is even closer for shorter intervals. Consider the ratio between the approximate and the true probabilities for an interval of length 0.0002: (dnorm(2.02,2,.24)*0.0002)/(pnorm(2.0201,2,.24) - pnorm(2.0199,2,.24)) # = 1 In fact, 1 - (dnorm(2.02,2,.24)*0.0002)/ (pnorm(2.0201,2,.24) - pnorm(2.0199,2,.24)) # = -2.873419e-08 Ted. -------------------------------------------------------------------- E-Mail: (Ted Harding) <Ted.Harding at manchester.ac.uk> Fax-to-email: +44 (0)870 094 0861 Date: 07-Sep-09 Time: 10:14:00 ------------------------------ XFMail ------------------------------
On Mon, Sep 7, 2009 at 12:42 AM, Gundala Viswanath<gundalav at gmail.com> wrote:> How do people usually use the result of density function (e.g. dnorm)? > Especially when its value can be greater than 1. > > What do they do with such density >1? > >> dnorm(2.02,2,.24) > [1] 1.656498There are countless uses. E.g., when viewed as a function of the parameters it is a likelihood, allowing one to estimate parameters given data... hth, Kingsford> > > - G.V. > > ______________________________________________ > R-help at r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. >