PDFtest

Recommendations

Info

Definition. X is a continuous random variable if there is a function f(x) so that for any constants a and b, with −∞ ≤ a ≤ b ≤ ∞, P(a ≤ X ≤ b) = ∫ b a f(x) dx (1) • For δ small, P(a ≤ X ≤ a + δ) ≈ f(a) δ. • The function f(x) is called the probability density function (p.d.f.). • For any a, P(X = a) = P(a ≤ X ≤ a) = ∫ a f(x) dx = 0. a • A discrete random variable does not have a density function, since if a is a possible value of a discrete RV X, we have P(X = a) > 0. • Random variables can be partly continuous and partly discrete. 2
The following properties follow from the axioms: • ∫ ∞ f(x) dx = 1. −∞ • f(x) ≥ 0. Example. For some constant c, the random variable X has probability density function ⎧ ⎨ cx n 0 < x < 1 f(x) = ⎩ 0 otherwise Find (a) c and (b) P(X > x) for 0 < x < 1. 3
Page 1: 5. Continuous Random Variables Cont
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 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

PDFtest

Create successful ePaper yourself

Delete template?

Save as template?