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Afternoon.

I'm looking into using the Gamma (Erlang) distribution for a certain quantity that I need to model. I noticed by plugging in some values for the distribution parameters that the y axis values which represent the probability that a random value from the x axis be drawn (unless I've gotten it all wrong) can jump above the value of 1, which doesn't make any sense for a probability distribution.

For an example, check out this example distribution produced by Wolfram Alpha: http://www.wolframalpha.com/input/?i=Gamma%282%2C+0.1%29&a=*MC.Gamma%28-_*DistributionAbbreviation-

Obviously there's some misconception on my part here. Care to point it out for me? Thanks.

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If you are looking at the PDF in your (broken) link, it can indeed have values greater that 1. But evaluating the PDF at a point does not give you a probability. You obtain probabilities from the PDF by integration. For instance $P[a<x<b]$ is the integral of the PDF over $[a,b]$; or, the area under the graph of the PDF over $[a,b]$. It is these areas that are bounded by 1 –  David Mitra May 8 '12 at 16:47

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A probability density function can easily be greater than $1$ in some interval, as long as the total area under the curve is $1$.

As a simple example, suppose that the random variable $X$ has uniform distribution on the interval $[0,1/4]$. Then the appropriate density function is $f_X(x)=4$ on $[0,1/4]$, and $0$ elsewhere.

For a more complicated example, look at a normally distributed random variable with mean $0$ and standard deviation $1/10$. You can easily verify that the density function of $X$, for $x$ close to $0$, is greater than $1$.

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