Math Help Forum - View Single Post - findings mean, SD and approximate distribution

Random Variable · #4 June 30th, 2009, 10:28 AM

If $X_{1}, X_{2}, ... , X_{n}$ are indepedent random variables with respective means $\mu_{1}, \mu_{2}. ... , \mu_{n}$ and variances $\sigma^{2}_{1}, \sigma^{2}_{2}, ... , \sigma^{2}_{n}$

then the mean and variance of $Y = \sum^{n}_{i=1} a_{i} X_{i}$ are $\mu_{Y} = \sum^{n}_{i=1} a_{i} \mu_{i}$ and $\sigma^{2}_{Y} = \sum^{n}_{i=1} a_{i}^{2} \sigma^{2}_{i}$ respectively

You can prove the preceding directly from the definition.

Therefore, $\mu_{\bar{X}} = \sum^{n}_{i=1} \frac {1}{n} \mu = \frac {1}{n} (n \mu) = \mu$

and $\sigma^{2}_{\bar{x}} = \sum^{n}_{i=1} (\frac {1}{n})^{2} \sigma^{2} = \frac{1}{n^{2}} (n \sigma^{2}) = \frac {\sigma^{2}}{n}$

part(c) is the result of the central limit theorem