Counting Techniques

  1. Multiplication rule
    • Suppose there are $r$ items in a set
    • Also, let’s say there are $n_1$ possibilities for the $r=1^{\mathrm{st}}$ item, $n_2$ possibilities for the $r=2^{\mathrm{nd}}$ item, …, and $n_r$ possibilities for the $r=r^{\mathrm{th}}$ item
      • Then, the total number of possibilities for all the different $r$ items is $(n_1)(n_2) \cdot … \cdot (n_r)$
  2. Permulations 
    • permutation is an ordered set of $r$ items selected from a (larger) set of $n$ items without replacement
    • Note: these are a special case of the multiplication rule
      • Specifically, we use permutations when the order in which we choose an item from a set is relevant
    • Theorem: the number of $n$ distinct items selected $r$ at a time without replacement is… \begin{equation} \frac{n!}{(n-r)!} \end{equation}
  3. Combinations 
    • combination is an unordered set of $r$ items taken from a (larger) set of $n$ objects without replacement
    • Theorem: the number of combinations of $r$ items taken without replacement from a (larger) set of $n$ distinct objects is… \begin{equation} \binom{n}{r} = \frac{n!}{(n-r)!r!} \end{equation}

Random Variables

Two different kinds of random variables:

  1. Discrete random variables: characterized by a probability mass function
  2. Continuous random variables: characterized by a probability density function


**CORRECTION: As pointed out by adityaguharoy, there acutally random variables/distributions that are both discrete and continuous (e.g., the cumulative distribution function). Hopefully I’ll remember update this post with more details 🙂

3 thoughts on “Probability”

  1. Errata : Actually there are Random Variables which are neither continuous nor discrete. We can construct such random variables by simply making the cumulative distribution function non continuous at at least one point but keeping the range of the Random Variable uncountable.

    1. I know I learned about the cumulative distribution function in prob/stats, but I just don’t have my notes on me. I’ll try and edit this in the near future. Thanks for the correction! (and sorry for responding so late XP)

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