# 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] = {t: 0 ≤ y ≤ T}. |

**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:

{AB: A and B are points on a given circle}. |

One natural random variable defined on this space is the length of the chord. The variable takes on random length on the interval

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.,

On a continuous space we consider a non-negative function, say, f(t) - called *probability density* - that satisfies

∫f(t)dt = 1, |

where the integration is over the interval *uniform*.) Then, for any subinterval

P(a, b) = P([a, b]) = (b - a) / T. |

In particular,

P(0, T) = (T - 0) / T = 1, |

as expected.

Along with f we usually define the *probability distribution* function F as the probability from the beginning of the interval of definition:

F(t) = (t - 0) / T = t/T, |

or, setting in a more general case

F(t) = ∫f(s)ds, |

where now the integral is taken over interval

F(0) = 0 and F(T) = 1. |

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