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The Standard Normal Distribution Definition: Z has the standard normal distribution if it has p.d.f. f(x) = 1 √ 2π e −x2 /2 • f(x) is symmetric about x = 0, so E(X) = 0. • Var(X) = 1. Check this, using integration by parts: 18
The Normal Distribution Definition. If X has density f(x) = { } 1 −(x − µ) 2 √ exp 2πσ 2 2σ 2 then X has the normal distribution, and we write X ∼ N(µ, σ 2 ). • E(X) = µ and Var(X) = σ 2 . • If Z ∼ N(0, 1) then Z is standard normal. Proposition. Let Z ∼ N(0, 1) and set X = µ + σZ for constants µ and σ. Then, X ∼ N(µ, σ 2 ). Proof. 19
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: Discussion problem. Suppose X ∼ U
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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