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Using the probability distribution density functions dbinom, dnorm, etc. and the corresponding cumulative probability functions pbinom and pnorm, I noticed that the dnorm density values could be smaller than the corresponding cumulative probability pnorm values. In the example below, the dnorm value for an x=5.2 is about 5 times larger (.005) than the corresponding pnorm value (.001). How can this be?

x <- seq(4, 10, .3)
point_density <- round(dnorm(x, 6.8, .5),3)
cumulated_probability <- round(pnorm(x, 6.8, .5),3)
df <- data.frame(x=x, point_density, cumulated_probability)
head(df,20)
jbowman
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alwinhoff
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    `dnorm` is the derivative of `pnorm` at `x`; there's no reason at all why the derivative of a function can't be smaller than, equal to, or larger than the value of that function at a particular point. – jbowman Oct 08 '18 at 17:04
  • @jbowman, this should be an answer. If you add a cpuple of concrete examples illustrating your point, I think thats is all that is needed. – Richard Hardy Oct 08 '18 at 17:12
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    @RichardHardy - have done, thanks for the suggestion. I know I'm not supposed to answer questions in comments, but sometimes I do it anyway if it's just a couple of sentences - a habit I'll try to break. – jbowman Oct 08 '18 at 17:23
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    I don't think it's a vice to give brief answers in comments. It's a natural thing to do if you want to say something informative and do that promptly but don't have the time and inclination to expand on it just now. Someone else may post a fuller answer very soon. If not, then the encouragement to expand on it is always welcome. So everyone can be right. – Nick Cox Oct 08 '18 at 18:05
  • @NickCox, I agree. – Richard Hardy Oct 08 '18 at 18:48

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dnorm is the derivative of pnorm at x; there's no reason at all why the derivative of a function can't be smaller than, equal to, or larger than the value of that function at a particular point.

Let's look at a plot of the standard Normal distribution (dnorm) and its cumulative density function (`pnorm'):

enter image description here

where clearly dnorm(x) > pnorm(x) for a considerable part of the plotted domain of x.

jbowman
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    This is correct in terms of the numbers emitted by such functions, which are for dimensionless arguments. However, in general, functions and their derivatives have different units and dimensions, so they can't even be compared. You can't even compare 1 km/hr with 10 km to say which is greater or lesser or whether they are equal. – Nick Cox Oct 08 '18 at 18:01