Module4.pdf - Joint Marginal\u2014 4.1,4.3 Cov Corr\u2019n \u2014 4.2 Conditional Ds\u2019ns 4.3 Bivariate Normal \u2014 4.4 Methods of Mathematical Statistics 4

Module4.pdf - Joint Marginal— 4.1,4.3 Cov Corr’n —...

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Joint, Marginal— 4.1,4.3 Cov, Corr’n — 4.2 Conditional Ds’ns - 4.3 Bivariate Normal — 4.4 Methods of Mathematical Statistics 4 Bivariate Distributions Tim Brown and Guoqi Qian Methods of Math. Stats.: Bivariate 1/51
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Joint, Marginal— 4.1,4.3 Cov, Corr’n — 4.2 Conditional Ds’ns - 4.3 Bivariate Normal — 4.4 Where have we come from? PMF/PDF For any real x the Probability Mass/Density Function f ( x ) , is given by f ( x ) = P ( X = x ) , X discrete F ( x ) , X continuous, F the CDF (1) Methods of Math. Stats.: Bivariate 2/51
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Joint, Marginal— 4.1,4.3 Cov, Corr’n — 4.2 Conditional Ds’ns - 4.3 Bivariate Normal — 4.4 Where have we come from? PMF/PDF For any real x the Probability Mass/Density Function f ( x ) , is given by f ( x ) = P ( X = x ) , X discrete F ( x ) , X continuous, F the CDF (1) Properties (a) 0 f ( x )( and 1 for X discrete ) and f ( x ) > 0 on the possible values of X (b) x range ( X ) f ( x ) = 1 , X discrete -∞ f ( x ) dx = 1 , X continuous (c) for any subset A of reals P ( X A ) = x A f ( x ) , X discrete A f ( x ) dx, X continuous Methods of Math. Stats.: Bivariate 2/51
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Joint, Marginal— 4.1,4.3 Cov, Corr’n — 4.2 Conditional Ds’ns - 4.3 Bivariate Normal — 4.4 Where are we going? Consider two random variables X, Y rather than one. Methods of Math. Stats.: Bivariate 3/51
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Joint, Marginal— 4.1,4.3 Cov, Corr’n — 4.2 Conditional Ds’ns - 4.3 Bivariate Normal — 4.4 Where are we going? Consider two random variables X, Y rather than one. Assume both are discrete or both are continuous. Methods of Math. Stats.: Bivariate 3/51
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Joint, Marginal— 4.1,4.3 Cov, Corr’n — 4.2 Conditional Ds’ns - 4.3 Bivariate Normal — 4.4 Where are we going? Consider two random variables X, Y rather than one. Assume both are discrete or both are continuous. Base our calculations to do with X, Y together on a joint PMF/PDF. Methods of Math. Stats.: Bivariate 3/51
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Joint, Marginal— 4.1,4.3 Cov, Corr’n — 4.2 Conditional Ds’ns - 4.3 Bivariate Normal — 4.4 Where are we going? Consider two random variables X, Y rather than one. Assume both are discrete or both are continuous. Base our calculations to do with X, Y together on a joint PMF/PDF. Aim: Extend our capability to deal with independent RVs (Variance, MGF) to dependent RVs. Methods of Math. Stats.: Bivariate 3/51
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Joint, Marginal— 4.1,4.3 Cov, Corr’n — 4.2 Conditional Ds’ns - 4.3 Bivariate Normal — 4.4 Where are we going? Consider two random variables X, Y rather than one. Assume both are discrete or both are continuous. Base our calculations to do with X, Y together on a joint PMF/PDF. Aim: Extend our capability to deal with independent RVs (Variance, MGF) to dependent RVs. Need to consider joint probabilities for X, Y . Methods of Math. Stats.: Bivariate 3/51
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Joint, Marginal— 4.1,4.3 Cov, Corr’n — 4.2 Conditional Ds’ns - 4.3 Bivariate Normal — 4.4 New Notation for two random variables X, Y Write for any sets A, B of real numbers the event [ X A ] [ Y B ] as [ X A, Y B ] Methods of Math. Stats.: Bivariate 4/51
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Joint, Marginal— 4.1,4.3 Cov, Corr’n — 4.2 Conditional Ds’ns - 4.3 Bivariate Normal — 4.4 New Notation for two random variables X, Y Write for any sets A, B of real numbers the event [ X A ] [ Y B ] as [ X A, Y B ] Example [ X = 1] [ Y = 2] is written as [ X = 1 , Y = 2] Methods of Math. Stats.: Bivariate 4/51
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Joint, Marginal— 4.1,4.3 Cov, Corr’n — 4.2 Conditional Ds’ns - 4.3 Bivariate Normal — 4.4
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