Math 311

Class 3

Probability Model

  • Sample space \(S\): the set of all possible outcomes
  • Set of observable events: \(\mathcal{E}\)
  • Probability: a function from the \(\mathcal{E}\) to \([0,1]\)
    • \(\operatorname{P}(S) = 1\)
    • For any sequence of events \(A_i\), \(i = 1, 2, 3 \ldots\) that are mutually exclusive, \(\displaystyle\operatorname{P}\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \operatorname{P}\left(A_i\right)\)
  • Addition rule: \(\operatorname{P}(A \text{ or } B) = \operatorname{P}(A) + \operatorname{P}(B) - \operatorname{P}(A\text{ and }B)\)
  • Conditional probability: \(\displaystyle\operatorname{P}(B\mid A) = \frac{\operatorname{P}(AB)}{\operatorname{P}(A)}\)
  • Multiplication rule: \(\operatorname{P}(AB) = \operatorname{P}(A)\times \operatorname{P}(B \mid A)\)
  • Independent events: \(\operatorname{P}(AB) = \operatorname{P}(A)\times \operatorname{P}(B)\)

Random Variables

  • \(X\colon S \to \mathbb{R}\) such that for each real number \(x\), \[\left\{s\in S \vert X(s) \le x\right\} \in \mathcal{E}\]
  • Vector random variable \(X \colon S \to \mathbb{R}^n\)
    • The sum of numbers on two dice.
    • The number of sixes when rolling two dice.
    • The (two-dimensional) vector of the two numbers on the two dice.
    • Number of “heads” when flipping a single coin.
    • The “time” we have to wait for something to happen.
    • The \(y\)-coordinate of the point where a photon strikes a square sensor.
    • Height of a randomly selected plant from a field.

Possible Values

  • The sum of numbers on two dice.
  • The number of sixes when rolling two dice.
  • The (two-dimensional) vector of the two numbers on the two dice.
  • Number of “heads” when flipping a single coin.
    • The “time” we have to wait for something to happen.
    • The \(y\)-coordinate of the point where a photon strikes a square sensor.
  • Height of a randomly selected plant from a field.
  • Discrete variable: finitely many or countably many possible values
  • Continuous variable: uncountably many possible values, with each single value having probability 0.

Cumulative Distribution Function

\[F(x) = \operatorname{P}\left(\left\{s \in S \vert X(s) \le x \right\}\right) = \operatorname{P}(X \le x)\]

  • Domain: \(\mathbb R\).
  • Range: \([0,1]\).
  • Nondecreasing: if \(a < b\) then \(F(a) \le F(b)\).
  • \(\displaystyle\lim_{x\to-\infty} F(x) = 0\).
  • \(\displaystyle\lim_{x\to\infty} F(x) = 1\).
  • \(\operatorname{P}(a < X \le b) = F(b) - F(a)\).

Discrete Variables

  • Finitely many or countably many possible values

  • CDF is “piecewise constant” (step functions or “jump” function)

  • The “jumps” occur exactly at the possible values.

  • The height of each “jump” is exactly the probability of that value.

  • Probability Mass Function (PMF), sometimes called Probability distribution function (PDF) gives probability for each possible value.

    \(p(a) = \operatorname{P}(X = a)\)

  • Sum over all possible values must be 1.

  • \(\displaystyle F(x) = \sum_{x_i \le x} p(x_i)\)