lesson5 - Continuous probability distribution

lesson5 - Continuous probability distribution - Lesson5:

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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson5-1 Lesson 5: Continuous Probability  Distributions
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson5-2 Outline Continuous probability distributions Features of univariate probability distribution Features of bivariate probability distribution Marginal density and Conditional density Expectation, Variance, Covariance and Correlation Coefficient Importance of normal distribution The normal approximation to the binomial
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson5-3 Types of Probability Distributions Number of random variables Joint  distribution 1 Uni variate probability distribution 2 Bi variate probability distribution 3 Tri variate probability distribution n Multi variate probability distribution Probability distribution may be classified according to the  number of random variables it describes.
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson5-4 Continuous Probability Distributions continuous random variable  is a variable that can assume  any value in an interval thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on  the ability to measure accurately.
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson5-5 Cumulative Distribution Function The  cumulative distribution function , F(x), for a continuous  random variable  X  expresses the probability that  X  does not  exceed the value of  x Let  a  and  b  be two possible values of  X, with  a < b.  The  probability that  X  lies between  a  and  b  is x) P(X F(x) = F(a) F(b) b) X P(a - = < <
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson5-6 0 0.2 0.4 0.6 0.8 1 1.2 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Number of heads (X) Cumulative probability Probability mass Example Discrete probability distribution x) P(X F(x) = X: the number of heads in an experiment of tossing three coins
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson5-7 Example Uniform distribution over the interval (0,1) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.25 0 0.25 0.5 0.75 1 1.25 U(0,1) density U(0,1) cumulative distribution x) P(X F(x) =
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson5-8 Example Standard normal distribution 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 -3 -2 -1 0 1 2 3 4 x Standard normal density Standard normal cumulative probability distribution x) P(X F(x) =
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson5-9 Probability Density Function Let X be a random variable that takes any real values in an  interval between  a  and  b .  The number of possible outcomes are  by definition infinite.   The main features of a probability density function f(x) are: P(X   (- , + )) = P(X   (a,b)) = 1.
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lesson5 - Continuous probability distribution - Lesson5:

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