Sampling Distributions

Normal distribution results
Consider $X_1,\dots ,X_n\sim N(\mu ,{\sigma }^2)$. Then if $\overline{X}=\sum X_i/n$ and $S^2=\sum (X_i-\overline{X}{)}^2/(n-1):$

1. $\overline{X}\sim N(\mu ,{\sigma }^2/n)$
2. $\sum (X_i-\mu {)}^2/{\sigma }^2\sim {\chi }_n^2$
3. $\sum (X_i-\overline{X}{)}^2/{\sigma }^2\sim {\chi }_{n-1}^2$
4. $(n-1)S^2/{\sigma }^2\sim {\chi }_{n-1}^2$
5. $\frac{\overline{X}-\mu }{S/\sqrt{n}}\sim t_{n-1}$

• fundamental to statistical inference
  • relating statistics from the sample to the population parameters
  • we use arguments based on sampling distribution of the statistic to state how close the estimate is likely to be to the parameter
    • sampling distribution of a statistic is the probability distribution of that statistic
      • = probability distributions if samples of the same size were to be repeatedly drawn from the population
        • but we dont really do it, math can figure it out
• sampling distributions of
  • $\overline{X}$
    • 
  • $S^2$
    • 
• sample distribution of the sample mean($\overline{x}$)
  • assumed that population is $\mathrm{\infty }$ or very large compared to sample size
    • otherwise math changes a little
  • $\overline{x}$
    • $X_1,X_2,...,X_n$ are n independently drawn observations from a population with mean $\mu$ and sd $\sigma$
    • $\overline{X}=\frac{X_1+X+2+...+X_n}{n}$
      • n = sample size, not amount of sample draws
      • $E(\overline{X})=E(\frac{X_1+X_2+...+X_n}{n})=\mu$
        • $E(X_1)=E(X_2)=E(X_n)=\mu$
          • thats how expectation works
        • $\overline{X}$ is an unbiased estimate of $\mu$
        • the expectation of a sum is always the sum of the expectations
        • we sometimes write $E(\overline{X})=\mu$ as ${\mu }_{\overline{X}}=\mu$
          • mean of the sampling distribution of the sample mean = mean of the population
            • $\overline{X}$ can take multiple values in different samples, but their mean will be equal to the population mean
          • mean of $\overline{X}$, the random variable, same as expectation
      • $Var(\overline{X})=Var(\frac{X_1+X_2+...+X_n}{n})=\frac{{\sigma }^2}{n}$
        • if the random variables are independent, the variance of their sum is the sum of their variances
        • often denoted as $\text{sigma\_{bar X}^2 = frac{sigma { <a class="tag" onclick="toggleTagSearch(this)" data-content="\#2">\#2</a>} }{n}}$
          • $\sigma (\overline{X})=\frac{\sigma }{\sqrt{n}}$
            • sd of the mean of multiple samples taken will have a smaller variance
      • $\overline{X}$ is normally distributed if we are sampling from a normally distributed population
      • $\overline{X}\sim N(\mu ,\frac{{\sigma }^2}{n})$
        • for the mean of n observations
        • $Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}$
          • tends to $Z\sim N(0,1)$ as $n\to \mathrm{\infty }$
            • tends to the standardized normal distribution
  • ${\mu }_{\overline{x}}$ = mean of $\overline{x}$
  • ${\sigma }_{\overline{x}}$ = standard deviation of $\overline{x}$

If $S_1^2$ and $S_2^2$ are the sample variances of two random samples of size $n_1$ and $n_2$ from two populations with the variances ${\sigma }_1^2$ and ${\sigma }_2^2$, then