The Strong Law of Large Numbers

Title

Context:

• Source:
• Relation:

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Insight:

• average of a sequence of independent RV have a common distribution will, with probability 1, converge to the mean of that distribution
• The Strong Law of Large Numbers
  • $X_1,X_2,\cdots ,$ are sequence of independent + identically distributed RV, each have finite mean $\mu =E[X_i]$. Then, with probability 1, $\frac{X_1+X_2+\cdots +X_n}{n}\to \mu \mathrm{as}n\to \infty$
  • Apply: use z-test to identify probability for each $a$, $P\{\frac{X-E[X]}{\sqrt{\mathrm{Var}(X)}}\le a\}\to \Phi (a)$