FunctionsRV - Le arning Obje s ctive C alculatefunctions of...

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Learning Objectives Calculate functions of discrete random variables. Define expected value and variance (both discrete & continuous) Calculate the expected value and variance of discrete random variables Calculate the expected value and variance of functions of discrete random variables. State and use the relationship between the mean and variance of one or more initial random variables and the mean and variance of a new random variable formed from a linear function of the initial random
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Example In your research on chronic bronchitis you plan to count the number of children who develop chronic bronchitis in the first year of life in 6 single-child households. If the probability for any one household is 5%, what is the expected number of children with chronic bronchitis?
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Functions of RVs A function of a discrete random variable defines another discrete random variable The PMF of the new RV can be obtained from the PMF of the original RV
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Functions of RVs Y = g ( X ) Old New RV X
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Example The RV X is the temperature in Celsius Transform to Fahrenheit by: Y = 1 .8 X + 32 Y = g ( X )
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Functions of RVs A function of a discrete random variable defines another discrete random variable The PMF of the new RV can be obtained from the PMF of the original RV
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Old New RV PMF Functions of RVs Y = g ( X ) p Y ( y ) = p X ( x ) { x | g ( x ) = y } p X ( x ) X
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Example Original RV X New RV Y = | X| p X ( x ) = 1/9 if x [ - 4,4] 0 otherw ise p Y ( y ) = 2/9 if y = 1,2,3,4 y = 0 0 otherwise X Y -4 4 -3 3 -2 2 -1 1 0 0 1 1 2 2 3 3 4 4 p Y ( y ) = p X ( x ) { x | g ( x ) = y }
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Example Original RV X New RV Y = X 2 p X ( x ) = 1/9 if x [ - 4,4] 0 otherw ise p Y ( y ) = 2/9 if y = 1,4,9,16 y = 0 0 otherwise X Y -4 16 -3 9 -2 4 -1 1 0 0 1 1 2 4 3 9 4 p Y ( y ) = p X ( x ) { x | g ( x ) = y }
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0 0.05 0.1 0.15 0.2 0.25 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 x or y p X ( x ) p Y ( y ) p X ( x ) = 1/9 if x [ - 4,4] 0 otherw ise p Y ( y ) = 2/9 if y = 1,4,9,16 y = 0 0 otherwise
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Example A family has 5 children obtained the old fashioned way and 2 adopted girls. What is the PMF of the number of girls out of the 7 children?
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This note was uploaded on 02/22/2010 for the course BME 335 taught by Professor Dunn during the Spring '10 term at University of Texas.

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FunctionsRV - Le arning Obje s ctive C alculatefunctions of...

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