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STAT 3743 Daily Outline

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2010-10-06 Wed

Quantile functions

Definition of quantile function:

Q(p) = min{x : F(x) >= p}, 0 < p < 1

Properties:

defined, finite for 0 < p < 1
left continuous
reflect F about line y=x (continuous case)
boundary limits exist but may be infinite

Special case: normal distribution

Define z_{α}

Examples

z_0.025, z_0.01, z_0.005

Functions of random variables

The idea

Discrete case

Make a table, accumulate probabilities

Example: X ~ binom, Y = something

PDF method

Introduce the formula and conditions

Intuitive formula

Example: X ~ norm, Y = linear transformation

Fact: if X is any norm then lin. trans. is also normal

Example: X ~ norm, Y = exp(X) (lognormal)

CDF method

Introduce the method

Example: X ~ unif(0,1), Y = -ln(X) (exponential)

Probability Integral Transform

Why the PIT is important

Author: G. Jay Kerns <gkerns@ysu.edu>

Date: 2010-10-08 15:21:17 EDT

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