STA256 Lecture 02

• Example 2:
  • $C_k=\{x:2<x\le 2+\frac{1}{k}\}$
  • Show this is decreasing
  • The left bound stays the same
  • Right bound, $\underset{k\to \mathrm{\infty }}{lim}2+\frac{1}{k}$
  • $\underset{k\to \mathrm{\infty }}{lim}2+0$
  • Eventually this gets to $2$, however the left bound means it has to be strictly greater than $2$, so it wouldn't work, but either way the right bound never touches $2$, it only approches it.
  • Notice that:
    • $2+\frac{1}{k}>2+\frac{1}{k+1}$
    • $\frac{1}{k}>\frac{1}{k+1}$
    • So this shows us that the right bound keeps on getting smaller, as the previous iteration's range is larger than the next iteration.
    • So we can conclude that this set $C_k$ is decreasing.
  • However $\underset{k\to \mathrm{\infty }}{lim}{C}_k=\mathrm{\varnothing }$ because we get $C_k=\{x:2<x<2\}$, so it's the $\mathrm{\varnothing }$.
  • Example to solve: $C_k=\{(x,y):\frac{1}{k}\le x^2+y^2\le 9-\frac{1}{k}\}$
    • Notice the lower bound $\underset{k\to \mathrm{\infty }}{lim}\frac{1}{k}=0$
    • Notice the upper bound $\underset{k\to \mathrm{\infty }}{lim}9-\frac{1}{k}=9$
    • So goes from lower $1\to 0$, and upper $8\to 9$
    • lower gets lower, upper gets upper
    • Increasing sequence
    • so $\underset{n\to \mathrm{\infty }}{lim}{C}_n=\{(x,y):0<x^2+y^2<9\}$
• Probability Set Function
• Experiment
• Random Experiment
• Sample Space
• Simple Event
• Compound Event
• Discrete Sample Space
• Practice Question:
  • Let sample space $S=\{1,2,3,4,5,6\}$
  • Let event $A=\{1,2,3,4\}$
  • Let $B=\{2,4,6\}$
• Probability Set Function
• Axioms of Probability