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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.
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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.
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The Normal Distribution
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The Normal Distribution
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The Multivariate 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 If variables with a multivariate normal distribution have covariances that equal zero, then the variables are independent.
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The Chi-Squared Distribution The chi-squared distribution is the distribution of the sum of m squared independent standard normal random variables.
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