Mathematical Expectation ENGR 0020: Probability and statistics for Engineers I
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This 3 page Class Notes was uploaded by Emily Binakonsky on Friday February 6, 2015. The Class Notes belongs to ENGR 0020: Probability and statistics for Engineers I at University of Pittsburgh taught by Maryam Mofrad in Spring2015. Since its upload, it has received 145 views. For similar materials see Probability and Statistics for Engineers 1 in Engineering and Tech at University of Pittsburgh.
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Date Created: 02/06/15
Mathematical Expectation Emily Binakonsky 1 Mathematical Expectation a Variance of Random Variables summarizes data well Indication of dispersion about the expected value Denotes by VarX or 039 Let X be the random variable with probability distribution f x and mean u The variance of X is given by If X is discrete 0 EX m 20c u2fx If X is continuous 00 a EX W f x u2fx dx 00 The standard deviation is the positive square root of the variance There is an alternative formula for finding variance It s 02 EX2 2 b The Variance of a Function Let X the random variable with probability distribution f x mean u The variance of the random variable gX is If X is discrete 09002 E 900 90602 2 2mm um fx x If X is continuous 09002 E 900 90602 f goo Hgx2fx dx 00 The standard deviation is the positive square root of the variance 0 Covariance of Two Random Variables Useful analysis of data to summarize the data Mathematical Expectation Emily Binakonsky Is the degree of association between the two variables in terms of deviation from their means If positive 0 Then when one random variable is larger than its mean the other random variable tends to be larger than its mean If negative 0 Then when one random variable is smaller than its mean the other random variable tends to be smaller than its mean Let X and Y random variables with joint probability distribution f x y The covariance of X and Y is If X and Y are discrete an E X x Y up i 2 22a M Y Myfx3I x 3 If X and Y are continuous oxy E X x y Hy f L X x Y Hyfxydxdy Another formula for Covariance of two random variables X and Y with means x and My respectively is given by KY 2 EXY x y d Correlation Coefficient Useful analysis tool to measure data s association Measures the degree of linearity between X and Y Denoted by pr Let X and Y random variables with covariance on and standard deviations ox and 03 respectively The correlation coef cient of X and Y is defined by pr XV UXUY Correlation Propositions 1 if a and c are either both positive or both negative paXbcYd pXY 2 for any two random variable s X and Y 1 S pr S 1 3 If X and Y are independent then p 0 but p 0 does not imply independence Mathematical Expectation Emily Binakonsky 6 Rules of the Expected Value If a and b are constants then EaX b aEX b Ifa 0 we will see that Eb b If 0 we will see that EaX aEX The expected value of the sum or difference of two or more functions of a random variable X is the sum or difference of the expected values of the functions That is to say EgX i W EgX i EhX The expected value of the sum or difference of two or more functions of the random variables X and Y is the sum or difference of the expected values of the functions That is to say EgXYihXY EgXY iEhXY f Independence Let X and Y two independent random variables Then EXY EXEY Let X and Y two independent random variables Then on 0 g Rules of Variance If X and Y are random variables with joint probability distribution f x y and a b and c are constants then anXerHc azoxz b2032 Zaboxy Set b 0 we will see Ua2Xc 120362 51202 Set a l and b 0 we will see 03 a 02 Set b 0 and c 0 we will see anX azoxz 2 1202 h Rules of Variance and Independence If X and Y are independent random variables then Oa2XbY 120362 1920312 If X and Y are independent random variables then oa2XbY a2 0x2 b2032 If X and Y are independent random variables then 2 2 2 2 2 u 2 2 Ua1x1a2x2anxn a1 0x1 a2 0x2 39 an Open