1.4: Continuous Probability Distributions
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Suppose 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 (1.2.4), (1.3.4), and (1.3.11) 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 and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)