Convergence

• If X is a nonnegative RV (support has no negative values) for which E(X) exists, then for t > 0
• 

• Let X be a RV with mean = µ and variance = σ², then for t > 0
• 

• 
• at least 75% of a RV distribution lies within two standard deviations of the mean

• 68
• 95
• 99.7

(Weak) Law of Large numbers

Central Limit Theorem

• Big picture goal is to estimate parameters such as µ
• If we get a RS we know that Y¯ will be a ‘close’ to µ for ‘large’ samples
• Applies to the average of any independent random variables with the same finite mean

Y_i \sim\limits^{\mathrm{iid}} \frac1n \sum_{i=1}^n Y_i \right\limits^{\mathrm{p}}E(Y)=\mu

The fundamentals of statistical inference!

What we are trying to do with inference is take sample data and relate it to a population with mathematical rigor

population mean over the sample size if you have a random sample