Blundell Thermal Physics · Problem 3.4

Probability

Physics — Statistical Mechanics · exponential-distribution

Problem

This question is about a continuous probability distribution known as the exponential distribution. Let xx be a continuous random variable that can take any value x≥0x \geq 0. A quantity is said to be exponentially distributed if it takes values between xx and x+dxx + \mathrm{d}x with probability P(x) dx=Ae−x/λ dx,P(x)\,\mathrm{d}x = A\mathrm{e}^{-x/\lambda}\,\mathrm{d}x, where λ\lambda and AA are constants. (a) Find the value of AA that makes P(x)P(x) a well-defined continuous probability distribution so that ∫0∞P(x) dx=1\int_0^{\infty} P(x)\,\mathrm{d}x = 1. (b) Show that the mean value of the probability distribution is ⟨x⟩=∫0∞xP(x) dx=λ\langle x \rangle = \displaystyle\int_0^{\infty} x P(x)\,\mathrm{d}x = \lambda. (c) Find the variance and standard deviation of this probability distribution. Both the exponential distribution and the Poisson distribution are used to describe similar processes, but for the exponential distribution xx is the actual time between, for example, successive radioactive decays, successive molecular collisions, or successive horse-kicking incidents (rather than, as with the Poisson distribution, xx being simply the number of such events in a specified interval).

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