2.5:Continuous Probability Distributions
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Suppose, now, that the variable u can take on a continuous range of possible values. In general, we expect the probability that u takes on a value in the range u to u+du to be directly proportional to du, in the limit that du→0. In other words,
P(u∈u:u+du)=P(u)du
where P(u) is known as the probability density. The earlier results (5), (12), and (19) generalize in a straightforward manner to give
1=∫∞−∞P(u)du⟨u⟩=∫∞−∞P(u)udu⟨(Δu)2⟩=∫∞−∞P(u)(u−⟨u⟩)2du=⟨u2⟩−⟨u⟩2
respectively.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)