(Solved): Suppose X is a random variable with mean X and variance X2=2. Suppose X1,X2, X3 ...

Suppose $$X$$ is a random variable with mean $${?}_X$$ and variance $${?}_X^2=2$$. Suppose $$X_1,X_2$$, $$X_3$$ and $$X_4$$ are all independently distributed with the same distribution as $$X$$. a) Calculate the variance of the sample mean $$\stackrel{?}{X}=\frac{X_1+X_2+X_3+X_4}{4}$$. b) Is the sample mean a better estimator of the population mean $${?}_X$$ than $$\tilde{X}=\frac{X_1}{3}+$$ $$\frac{X_2}{4}+\frac{X_3}{3}+\frac{X_4}{12}?$$ Justify your answer. c) You find out that $$X_1,X_2,X_3$$ and $$X_4$$ were gathered using a faulty method. $$X_1,X_2$$, $$X_3$$ and $$X_4$$ still have the same mean and variance as $$X$$, but $$X_2=X_3=X_4$$ because the lab assistant wrote the same observation 3 times. $$X_1$$ is still independently distributed. Given this mistake by the lab assistant, is the sample mean $$\stackrel{?}{X}=\frac{X_1+X_2+X_3+X_4}{4}$$ a better estimator of the population mean $${?}_X$$ than $$\tilde{X}=\frac{X_1}{3}+\frac{X_2}{4}+\frac{X_3}{3}+\frac{X_4}{12}?$$ Justify your answer.