ch2-2 - Chapter 2 Review of Probability (Part 2) The Normal...

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Chapter 2 Review of Probability (Part 2)

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The Normal Distribution A continuous random variable with a normal distribution has the bell-shaped probability density shown in Figure 2.5. I The normal distribution with mean μ and variance σ 2 is expressed as N ( μ , σ 2 ) . I It is symmetric around its mean and has 95% of its probability between μ 1.96 σ and μ + 1.96 σ . The normal distribution with mean μ = 0 and variance σ 2 = 1 is called the standard normal distribution . I Random variables that have a standard normal distribution are often denoted by Z . I The standard normal cumulative distribution function is denoted by Φ : Pr ( Z ± c ) = Φ ( c ) , where c is a constant. I Values of the standard normal cumulative distribution function are in Table 1.
The Normal Distribution

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The Normal Distribution Suppose Y is distributed N ( 1, 4 ) , what is the probability that Y 2? (What is the shaded area in Figure 2.6a?) To compute the probability, we standardize the variable Y by: I First, subtracting the mean, I and then dividing by the standard deviation. I ( Y ± 1 ) / p 4 = 1 2 ( Y ± 1 ) I Then the random variable 1 2 ( Y ± 1 ) is distributed N ( 0, 1 ) , standard normal (Figure 2.6b). Now Y 2 is equivalent to 1 2 ( Y ± 1 ) 1 2 ( 2 ± 1 ) , 1 2 ( Y ± 1 ) 1 2 : Pr ( Y 2 ) = Pr [ 1 2 ( Y ± 1 ) 1 2 ] = Pr ( Z 1 2 ) = Φ ( 0.5 ) = 0.691. I The skewness and the kurtosis of normal distribution are zero and 3, respectively.
The Normal Distribution

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The Normal Distribution
The joint distribution of a set of random variables is called the multivariate normal distribution . I If only two variables, it is called the bivariate normal distribution. The multivariate distribution has three important properties: I If n random variables have a multivariate normal distribution, then any linear combination of these variables is normally distributed. I If a set of variables has a multivariate normal distribution then the marginal distribution of each of the variables is normal. I

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This note was uploaded on 11/20/2011 for the course ECONOMICS 220:322 taught by Professor Otusbo during the Fall '10 term at Rutgers.

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ch2-2 - Chapter 2 Review of Probability (Part 2) The Normal...

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