Question

4 and 5

samples, the other in small samples. Which is which? Explain. (d) Suppose we know that the 5 values are from a symmetric distribution. Then the sample median is also unbiased and consistent for the population mean. The sample mean has lower variance. Would you prefer to use the sample 4. Suppose Yi, Y, are iid r ables with E(n)-μ, Var(K)-σ2 < oo. For large n, find the approximate 5. Suppose we observe Yi...Yn from a normal distribution with unknown parameters such that Y 20, 216, and mean or sample median as an estimator of the population mean? Why? andom vari distribution of P-1 Σ, Yi, Be sure to name any theorems you used. n = 10.

Answer 1

4) Here Y₁ , Y₂ , ..., Y_n are iid random variables.

For large n,

$\bar Y =\frac{1}{n}\sum_{i=1}^{n}Y_{i}$

which is a sample mean. Also Y_i is a random variable with finite variance.

Therefore, by central limit theorem $\bar Y$ follows normal distribution with mean = $\mu$ and variance = $\sigma ^{2}/n$ .