Notethetotalareaunderthepdfcurveisequaltoone 10

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: X ) = 0 ⎣ σX ⎦ X ⎡ X − μX ⎤ 1 = 2 Var(X ) = 1 Var(Z) = Var ⎢ ⎥ ⎣ σX ⎦ σX and That is, the standardized random variable Z has mean 0 and variance 1. 9 Chapter 6 Chapter 6.3 The Normal Distribution The continuous random variable that follows the normal distribution has some popularity in applied work. The probability density function (PDF) for a normally distributed random variable X with mean μ X and variance σ X is: 2 f( x ) = ⎛ ⎞ 1 exp⎜ − (x − μ X )2 ⎟ for − ∞ < x < ∞ ⎜ 2 σ2 ⎟ 2 π σ2 X ⎝ ⎠ X 1 f(x) Graph of the PDF for a Normal Distribution E(X) x The shape of the PDF is a symmetric, bell‐shaped curve centered on the mean μ X . Note: the total area under the PDF curve is equal to one. 10 Chapter 6 To state that a random variable X follows a normal distribution summarized by the parameters mean μ X and variance σ X the 2 notation is: X ~ N(μ X , σ 2 ) X ↑ “is distributed as” The cumulative distribution function (CDF) is: F(x) = P(X ≤ x ) The graph shows that the area under the PDF to the left of the value a is the cumulative probability F(a ) . f(x) Area = P(X < a) = F(a) E(X) x 11 a Chapter 6 For two values a and b the range probability is calculated from the CDF as: P(a < X < b) = F(b) − F(a ) The next graph shows that the area under the PDF between the values a and b is the range probability. f(x) Area = P(a < X < b) a...
View Full Document

This note was uploaded on 02/06/2014 for the course ECON ECON 325 taught by Professor Whistler during the Spring '10 term at The University of British Columbia.

Ask a homework question - tutors are online