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Proposition 2. For X a continuous RV with p.d.f. f and any real-valued function g Proof. E [g(X)] = ∫ ∞ −∞ g(x)f(x)dx 10
Example. A stick of length 1 is split at a point U that is uniformly distributed over (0, 1). Determine the expected length of the piece that contains the point p, for 0 ≤ p ≤ 1. Solution. 11
Page 1 and 2: 5. Continuous Random Variables Cont
Page 3 and 4: The following properties follow fro
Page 5 and 6: Cumulative Distribution Function (c
Page 7 and 8: Example. Suppose X has p.d.f. ⎧
Page 9: Proposition 1. For X a non-negative
Page 13 and 14: The Uniform Distribution Definition
Page 15 and 16: Example. Let X ∼ U[a, b]. Show th
Page 17 and 18: Discussion problem. Suppose X ∼ U
Page 19 and 20: The Normal Distribution Definition.
Page 21 and 22: Calculating with the Normal Distrib
Page 23 and 24: Example continued. For what value o
Page 25 and 26: The Normal Approximation to the Bin
Page 27 and 28: Example. The ideal size of a first-
Page 29 and 30: Example. The amount of time, in hou
Page 31 and 32: Example. At a certain bank, the amo
Page 33 and 34: • Calculations similar to the pre

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