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Basics of Sampling Theory

Basics of Sampling Theory. P = { x 1 , x 2 , ……, x N } where P = population x 1 , x 2 , ……, x N are real numbers Assuming x is a random variable; Mean/Average of x ,. N. Σ x i x = N. i=1. Basics of Sampling Theory. Standard Deviation , Variance ,. N. √. Σ (x i – x) 2

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Basics of Sampling Theory

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  1. 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 = N i=1

  2. Basics of Sampling Theory Standard Deviation, Variance, N √ Σ (xi – x)2 σx = N i=1 N Σ (xi – x)2 σx2 = N i=1

  3. 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

  4. 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

  5. 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 * Error depends on number of samples; bigger sample – less error; smaller sample – more error * This formula is true for sampling with replacement * This theory works only on independent variables; while mean theorem works on dependent variable

  6. Basics of Sampling Theory Normal Distribution curve σ x2 xN x1 x Assuming numbers are sorted • K = number of samples • z = sample mean • as k increases, z comes closer to x

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