Question

For the same topic

5) Consider an i.i.d. population {X1, X2,...} and take a sample of size 4. Show that if we take ſ = X 4 to be a point estimator for population mean, that this is an unbiased estimator. 6) Consider another i.i.d. population, {X1, X2,...}, and take o2 = X to be an estimator for population variance. Find the bias in this estimator Is it unbiased? If not, does it give values that are too high or too low, on average?

Answer 1

Pprarionxi, Here we are gven and vonianec 62 with population meon Now wo are to take a somple of size Ru- We ape to shous that is an unbiased cetmotoe Poo enimating ASe popularion mcon 1e! =(G) S*CH) 4 E C As xi's are id Pondom eyvorlables 4 4 Ju for X is an unloiased estimatop e atimating

Unbiased estimator=The unbiased estimator is a accurate statistic from sample to estimate the population parameter,here accuracy means the statistic give value of the population parameter neither overestimated nor underestimated.More mathematically,E($\hat{\Theta }$)=$\Theta$,if a estimator is biased then the bias would be calculated by the mean of the difference of the estimator and the parameter,more mathematically E($\hat{\Theta }$-$\Theta$)=BIAS.

Now are to shouw that unbiased ostimaron a ore T eatimating the Populanon vaniamee 62 or not e. E(x)vCxLE(xy the den of the varionee d anv -63 Cetas Xis a blased ostimator As ue know X, is a nv. with mean d & vorlanee 6 with blas du As we knous that Fop blased eeton.artroo ewe Note Fs(T-rCo)-p()uhere ce i Parametoic Funetion Ard P(e) is a bias. 3o X is not unblosed Aa X is not an unblased cstimaton mating ు ల ape to 4hink af glues ee too lous on aep values tbo that eithee As whatever the meam (is negative poctve) the squane OP f it is aluys X givG over estimatod value of me paromat e, Henee Xve values too high, on aveHage.

A SMALL CORRECTION IN THE ABOVE IMAGE:

[X₁² IS A BIASED ESTIMATOR WITH BIAS $\mu$ ²]

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