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Exercise. Check that the standard normal density integrates to 1. Trick. By changing to polar coordinates, show ( ∫ ∞ 2 exp{−x 2 /2} dx) = 2π. −∞ Solution. 20
Calculating with the Normal Distribution • There is no closed form solution to the integral √1 2π e −x2 /2 dx, so we rely upon computers (or ∫ b a tables). • The c.d.f. of the standard normal distribution is ∫ z 1 Φ(z) = √ e −x2 /2 dx. 2π −∞ This is tabulated on page 201 of Ross. Example. Find ∫ 2 −1 Solution. 1 √ 2π e −x2 /2 dx. 21
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 and 10: Proposition 1. For X a non-negative
Page 11 and 12: Example. A stick of length 1 is spl
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: The Normal Distribution Definition.
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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