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The Probability Density Function

If you really feel you must have some sort of mathematical explanation, try this for size:

Mathematically, the normal distribution is defined so that the probability density function is -
P(x) = 1 e
 
-(x - m)2 / 2 σ2
σ√(2π)

Being a proportion and a probability, the area under the curve P(x) describes adds up to one. Which is why, as the standard deviation becomes smaller, the peak becomes narrower and higher. For example, if the standard deviation is 0.1, Zmax is 39.9

Clearly this is impossible if Z was a probability or a proportion, and confuses many people. Let us try to clarify this a little by calling 'Z', our probability density function, P(x) instead.

 

The first key point is that P(x) is a density, not a pure probability.

This means that if x has units of years or (metres, or kilograms), say, then P(x) is the probability per year (or metre or kilogram) that a certain result is found. The probability that we get a result in the small interval x to x + dx, where x could be 2 metres and dx could be 0.1 metres, that is a result between 2 and 2.1 metres, is:

P(x to x+dx) = P(x)dx

This result follows providing dx is small enough for P(x) to be treated as constant over the interval x to x+dx.

Between -∞ and +∞ (minus infinity and plus infinity) however, P(x) is bound to change. So if we want the probability of getting a result between -∞ and some number, for example length a, we have to integrate (sum) from -∞ to a so that C(a), the cumulative distribution function, is:
C(a) = aP(x)dx
−∞

The point then is that C(a) is a pure probability, not a density. P(x) has units of 1/metres (that is a probability per meter) and dx has units of metres - so that P(x)dx is dimensionless.

 

The second key point is that C(∞) must be equal to 1, since it gives the probability that x lies between -∞ and +∞.

The factor 1/σ√(2π) is chosen to ensure that C(∞) = 1.

What this means, of course, is that the area under the curve of P(x) is one. So that, if the standard deviation is very small, then the peak of the curve must be very high (since it is the area that we are interested in).

Suppose then that σ = 0.001 metres, so that the peak value of P(x) is 1/0.001√(2π) = 398.9. All that this is saying is that the probability that x is between 0 and 0.0001 is 398.9 0.0001 = 0.3989.

When we do these approximate calculations (multiplying P(x) by dx rather than integrating P(x) from x to x+dx) the important thing is that dx must be very much less than σ. This ensures that P(x) can be taken as constant between x to x+dx, so that the probability is always going to be less than 1.