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The Distribution of Sample Maximums as a Function of Sample Size

Not All Sample Statistics Approximate a Normal Distribution as Sample Size Increases

Consider the distribution of sample maximums for samples of a random variable uniformly
distributed between -0.5 and +0.5. For n=1 the sample maximum is just the sample value.

The above distributions indicate the necessity that for an extension of the central limit
theorem to apply, the sample statistic must be representable as a sum.

Analysis of Sample Maximum

If p(x) is the probability density function for a random variable x, let P(x) be the
cumulative probability function; i.e.,

P(x) = ∫_{-∞}^{x}p(z)dz.

The probability that the maximum of a sample of size n is x is given by

[P(x)]^{n-1}p(x)

This is the probability density function q(x) for the sample maximum. When p(x) is the
uniform density function

p(x) = 1 for -0.5≤x≤+0.5
p(x) = 0 for all other values of x

then P(x) = (x-(-0.5) = x+0.5 for -0.5≤x≤+0.5. Thus the probability density function
for the sample maximum is given by:

q(x) = 0 for x≤-0.5
q(x) = n(x+0.5)^{n-1} for -0.5≤x≤+0.5
q(x) = 0 for +0.5≤x

where the factor of n is to take into account the n different ways the n-1 factors below x and
the one value of x can be arranged.

To check that q(x) is a proper probability density function consider its integration over the
interval [-0.5,x];i.e.,

The value of Q(x) at x=0.5 should be 1.0 and indeed Q(0.5) = 1^{n} = 1.0.

As n increases without bound this distribution goes to a spike function at x=0.5. There is
thus no tendency for the distribution of sample maximums to approach a normal distribution.

The purpose of the above analysis was to establish that there are sample statistics whose distribution does
not approach a normal distribution as the sample size increases without bound. It was convenient for
achieving this purpose to use a uniform, bounded distribution for the random variable. For the case of an
unbounded distribution see Unbounded distribution. For an analysis of the general
case see Sample Maximum.