Question

Let Xi, ,Xio be i.i.d. random variables with mean 10, and variance 100. (a) Select all the options that are correct. Explain your answers briefly. i. X10 is a random variable. ii. The mean of X10 is exactly 10. iii. The mean of X10 is approximately 10. iv. The variance of X10 is exactly 100. v. The variance of Xio is exactly 1 vi. The variance of X1o is approximately 1. (b) Find the P10). Is it approximate or exact? (c) Find the PXI 12). Is it approximate or exact? (d) Find the PXo 9). Is it approximate or exact?

Answer 1

a) i. Correct. Here sample mean is a statistic. And sample mean is based on 10 random variables. If random variables change sample mean will also change for different variables.

a)iii. Correct. Here population mean is 10 therefore our sample mean is not exactly 10 but approximately equal to 10. Because our sample mean statistic is unbiased estimator of population parameter.

a) v. Correct. The variance of $\overline{x_{_{10}}}$ is exactly equal to 1.

var($\overline{x_{_{10}}}$) = var(x)/n² =100/10² =100/100 =1

(b) $\mathbb{P}\left ( \overline{x_{10}} \right \leq 10)=$ $\mathbb{P}\left ( \overline{x_{10}}-10{/10}\leq \left ( 10-10 \right )/10 \right )$^{$= 0.5$}

(c) $\mathbb{P}\left ( \overline{x_{10}}\leq 12 \right )= \mathbb{P}\left ( \overline{x_{10}}-10/10 \right \leq \left ( 12-10 \right )/10)= \mathbb{P}\left ( z\leq 0.2 \right )=$ $= 0.4207$

(d)$\mathbb{P}\left ( \overline{x_{10}}\geq 9 \right )= \mathbb{P}\left ( \overline{x_{10}}-10/10 \right \geq \left ( 9-10/10 \right ))= \mathbb{P}\left ( z\geq -0.1 \right )= 1-\mathbb{P}\left ( z\leq 0.1 \right )= 1-0.4602= 0.5398$