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# Download Basics of Sampling Theory

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Basics of Sampling Theory P = { x1, x2, ……, xN} where P = population x1, x2, ……, xN are real numbers Assuming x is a random variable; Mean/Average of x, N Σ xi x= i=1 N Basics of Sampling Theory Standard Deviation, √ σx = Variance, σx2 N Σ (xi – x)2 i=1 N N Σ (xi – x)2 = i=1 N Basics of Sampling Theory Theorem About Mean picking random numbers x, mean = x picking random numbers y, mean = y x=y Picking another number z, mean z = x = y z = c1x + c2y ; c1, c2 are constants z=x+y Basics of Sampling Theory Independence two events are independent if the occurrence of one of the events gives no information about whether or not the other event will occur; that is, the events have no influence on each other for example a, b and c are independent if: - a and b are independent; a and c are independent; and b and c are independent Basics of Sampling Theory Theorem About Variances/Sampling Theorem z = (x + y)/2; σz2 = ? σz2 < σx2 Taking, z = (x + y)/2 σz2 = (σx2 + σy2)/4 Taking k sample, z = (x + x’ + x’’ + …. + x’’…k)/k σz2 = (kσx2 )/k2 σz2 = σx2 )/k * This theory works only on independent variables; while mean theorem works on dependent variable * Error depends on number of samples; bigger sample – less error; smaller sample – more error * This formula is true for sampling with replacement Basics of Sampling Theory Normal Distribution curve σ x1 x2 • K = number of samples • z = sample mean xN x • as k increases, z comes closer to x Assuming numbers are sorted

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