1. Sampling with replacement method: In this case we put the each item back inte the pepulatiun after selecting it and taking a measurement. Se, the

Sampling with replacement method: In this case we put the each item back into the population after selecting it and taking a measurement. So, the

1. Sampling with replacement method: In this case we put the each item back intethe pepulatiun after selecting it and taking a measurement. Se, the with unit may appearfl. [1 2, ..., n. time in the sample. Let t,- be the number at times that the F” unit appearsin the sample. Then the sample mean can be written as 1 Ny=;;tiyi- Since the probability that the a“ unit is drawn is if at each draw, the randum vari—able t,- has a BEn,p] distributien with p = Th? 2' = 1.,2, ....,N. Sc, EEti] = up andVarfli) = np[1 — p). Alse? jeintljr the HS fellew a multinemial distributien. Therefore,Coma-Jig) = -fir- Using these results and fallcming the same pruceclure used in theclass for the withflut replacement case, Shfl'ii’ that 1 N -Vii) = E Etta- - If)”.