STATSLIDE4 - PROBABILITY DISTRIBUTIONS: (continued) The...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: PROBABILITY DISTRIBUTIONS: (continued) The change of variables technique. Let x ∼ f ( x ) and let y = y ( x ) be a monotonic transformation of x such that x = x ( y ) exists. Let A be an event defined in terms of x , and let B be the equivalent event defined in terms of y such that if x ∈ A , then y = y ( x ) ∈ B and vice versa. Then, P ( A ) = P ( B ) and we can find the the p.d.f of y denoted by g ( y ). The continuous case. If x,y are continuous random variables, then Z y ∈ B g ( y ) dy = Z x ∈ A f ( x ) dx. If we write x = x ( y ) in the second integral, then the change of variable technique gives Z y ∈ B g ( y ) dy = Z x ∈ B f { x ( y ) } dx dy dy. If y = y ( x ) is a monotonically decreasing transformation, then dx/dy < 0, and f { x ( y ) } > 0, and so f { x ( y ) } dx/dy < 0 cannot represent a p.d.f since g ( y ) ≥ 0 is necessary. The recourse is to change the sign on the y-axis. When dx/dy < 0, it is replaced by its modulus | dx/dy | > 0. In general, g ( y ) = f { x ( y ) } Ø Ø Ø Ø dx dy Ø Ø Ø Ø . 1 The Standard Normal Distribution. Consider the function g ( z ) = e − z 2 / 2 , where −∞ < z < ∞ . There is g ( z ) > 0 for all z and also Z e − z 2 / 2 dz = 1 √ 2 π . It follows that f ( z ) = 1 √ 2 π e − z 2 / 2 constitutes a p.d.f., described as the standard normal and denoted by N ( z ; 0 , 1). In general, the normal distribution is denoted by N ( x ; μ,σ 2 ); so, in this case, there are μ = 0 and σ 2 = 1. The General Normal Distribution. This can be derived via the change of variables technique. Let z ∼ N (0 , 1) = 1 √ 2 π e − z 2 / 2 = f ( z ) , and let y = zσ +...
View Full Document

This note was uploaded on 03/02/2012 for the course EC 2019 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.

Page1 / 9

STATSLIDE4 - PROBABILITY DISTRIBUTIONS: (continued) The...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online