Continuous Sample Spaces
Let's return to the couple of examples of continuous sample spaces we looked at the Sample Spaces page:
Arrival time. The experimental setting is a metro (underground) station where trains pass (ideally) with equal intervals. A person enters the station. The experiment is to note the time of arrival past the departure time of the last train. If T is the interval between two consecutive trains, then the sample space for the experiment is the interval [0, T], or
Chord length. Given a circle of radius R, the experiment is to randomly select a chord in that circle. There are many ways to accomplish such a selection. However the sample space is always the same:
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{AB: A and B are points on a given circle}.
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One natural random variable defined on this space is the length of the chord. The variable takes on random length on the interval [0, T], where T is the diameter of the circle at hand. The length of a chord AB is zero if the two points happen to coincide.
How do we define the probability of an arrival time t in the first experiment or the length, say, L in the second?
We need to have P(Ω) = 1, i.e., P([0, T]) = 1. On the other hand, in the first experiment, all points in the interval [0, T] seem to be equiprobable. And, since the sum of the probabilities P(t) must be 1, it looks like we have arrived at an impossible situation. If P(t) is non-zero, the sum of all probabilities will be infinite; if P(t) is 0, the sum will vanish as well. The apparent paradox is resolved by pointing out that the notion of the sum of a continuum of values is commonly replaced by an integral - the concept taught at the beginning Calculus courses. The probabilities on a continuous sample space should be defined somehow differently and not point-by-point.
On a continuous space we consider a non-negative function, say, f(t) - called probability density - that satisfies
where the integration is over the interval [0, T]. And instead of talking of the probability of individual points P(t) we are concerned with the probability of a point falling into a small interval around t: [t - δ, t + Δ] which is defined as the integral ∫f(t)dt, where now the integral is taken over the interval [t - δ, t + Δ]. In the case of the first experiment, it makes sense to define f(t) ≡ 1/T, for any t in [0, T]. (Such a distribution is called uniform.) Then, for any subinterval [a, b],
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P(a, b) = P([a, b]) = (b - a) / T.
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In particular,
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P(0, T) = (T - 0) / T = 1,
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as expected.
Along with f we usually define the probability distribution function F as the probability from the beginning of the interval of definition:
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F(t) = (t - 0) / T = t/T,
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or, setting in a more general case
where now the integral is taken over interval [0, t]. In any event, the probability distribution is a non-negative function that satisfies
Copyright © 1996-2008 Alexander Bogomolny
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