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According to the info in the following locations:

The probability density function for a left-truncated normal distribution should be:

$f(x) = {{1 \over \sigma}\phi({{x-\mu} \over \sigma})\over{1-\Phi({{a-\mu} \over \sigma})}}$

for all x >= a

However, I'm getting some weird results when I do that.

if I take $\sigma$ = 1.0 and $\mu$=0.0 and $a$=1.0, that gives:

$f(x) = {\phi(x) \over {1-\Phi(1)}}$

If I then plugin a number near the truncation point (say 1.1), I get:

$f(x) \approx {0.212 \over {0.159}} \approx 1.3$

Clearly that's wrong (the PDF should be in the range (0.0, 1.0) for all X).

Except, everywhere I look leads me to the definition of f above, so I can't figure out what I'm doing wrong.

Clearly though, I've gotten something wrong.

I know this is an embarrassingly simple question, but I'd appreciate any help you might be able to give. Do you know what I'm doing wrong?

Jeromy Anglim
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Scott Wisniewski
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    The PDF does not have to be in the range (0, 1), it just needs to integrate to 1 over its domain. For example, the continuous uniform distribution U(0, 1/2) has PDF $f(x)=2$. – MLaz Aug 19 '14 at 05:18
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    A common mistake. Density is not probability. The issue here effectively duplicates [this](http://stats.stackexchange.com/questions/4220/a-probability-distribution-value-exceeding-1-is-ok) or [this](http://stats.stackexchange.com/questions/9427/interpreting-gaussian-probabilities-greater-than-1) and a number of others. – Glen_b Aug 19 '14 at 05:26
  • It's the area under a range within its support that will be between 0 and 1. – Luca Aug 19 '14 at 07:45
  • Even an _untruncated_ normal density function can have value greater than $1$. Consider, for example, the value of the density at $0$ for a normal random variable with mean $0$ and standard deviation smaller than $\frac{1}{\sqrt{2\pi}}$. – Dilip Sarwate Aug 19 '14 at 12:37

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You're confusing the PDF and the CDF. The CDF is bounded by 1, the PDF is not. This link contains a graph that's close to the PDF that you're calculating (3rd page of the pdf)

http://pages.stern.nyu.edu/~wgreene/DiscreteChoice/Readings/Greene-Chapter-19.pdf

jimijimjim
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