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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...
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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.
 Spring '10
 WHISTLER

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