{"id":523,"date":"2024-03-05T19:18:12","date_gmt":"2024-03-05T19:18:12","guid":{"rendered":"https:\/\/neilfoxman.com\/?page_id=523"},"modified":"2024-03-05T21:23:54","modified_gmt":"2024-03-05T21:23:54","slug":"probability","status":"publish","type":"page","link":"https:\/\/neilfoxman.com\/?page_id=523","title":{"rendered":"Probability"},"content":{"rendered":"\n

<\/span>General<\/span><\/h2>\n\n\n\n

An outcome $s$ is in sample space $S$<\/p>\n\n\n\n

$$
\\begin{align}
s &\\in S \\\\
S &= \\set{s_0, s_1, s_2, \\dots}
\\end{align}
$$<\/p>\n\n\n\n

An event $E$ is a subset of samples in the sample space<\/p>\n\n\n\n

$$
\\begin{align}
E \\subset S
\\end{align}
$$<\/p>\n\n\n\n

A Probability is a mapping of an event to a number that indicates how likely that event is to occur. Some Axioms:<\/p>\n\n\n\n

    \n
  1. For any event $A$, $P(A) \\ge 0$<\/li>\n\n\n\n
  2. $P(S) = 1$<\/li>\n\n\n\n
  3. If events $A_1, A_2, \\dots$ are disjoint ($\\forall A_i, A_j : A_i \\cup A_j = \\emptyset$) then $P(A_1 \\cup A_2 \\cup \\dots) = P(A_1) + P(A_2) + \\dots $<\/li>\n\n\n\n
  4. <\/li>\n<\/ol>\n\n\n\n

    <\/span>Conditional Probability<\/span><\/h2>\n\n\n\n

    https:\/\/www.probabilitycourse.com\/chapter1\/1_4_0_conditional_probability.php<\/a><\/p>\n\n\n\n

    $$
    \\begin{align}
    P(A|B) &= \\frac{P(A \\cap B)}{P(B)}
    \\end{align}
    $$<\/p>\n\n\n\n

    Note that<\/p>\n\n\n\n

      \n
    1. $A \\cap B = 0 \\implies P(A|B) = 0$<\/li>\n\n\n\n
    2. $A \\subset B \\implies P(A \\cap B) = P(A) \\implies P(A|B) = \\frac{P(A)}{P(B)}$<\/li>\n\n\n\n
    3. $B \\subset A \\implies P(A \\cap B) = P(B) \\implies P(A|B) = 1$<\/li>\n<\/ol>\n\n\n\n

      <\/span>Chain Rule<\/span><\/h3>\n\n\n\n

      $$
      \\begin{align}
      P(A|B) &= \\frac{P(A \\cap B)}{P(B)} \\\\
      P(B|A) &= \\frac{P(A \\cap B)}{P(A)} \\\\
      \\\\
      P(A \\cap B) &= P(A|B) P(B) = P(B|A) P(A) \\\\
      \\end{align}
      $$<\/p>\n\n\n\n

      \"\"\n\t\t\t\n\t\t\t\t\n\t\t\t<\/svg>\n\t\t<\/button><\/figure>\n\n\n\n

      <\/span>Independence<\/span><\/h3>\n\n\n\n

      $$
      \\begin{align}
      A, B \\text{ are independent} &\\iff P(A \\cap B) = P(A) P(B) \\\\
      \\\\
      P(A|B) &= \\frac{P(A \\cap B)}{P(B)} \\\\
      P(A|B) &= \\frac{P(A)P(B)}{P(B)} \\\\
      P(A|B) &= P(A) \\\\
      \\end{align}
      $$<\/p>\n\n\n\n

      <\/span>Random Variables<\/span><\/h2>\n\n\n\n

      A random variable $X$ is a mapping from specific outcomes to the real number line<\/p>\n\n\n\n

      $$
      \\begin{align}
      X : S \\rightarrow \\mathbb{R}
      \\end{align}
      $$<\/p>\n\n\n\n

      Note that a random variable is a function, but is often treated\/notated as a scalar.<\/p>\n\n\n\n

      A random variable will have a Range, or a set of possible values for $X$.<\/p>\n\n\n\n

      $$
      \\begin{align}
      Range[X] &= R_X \\\\
      R_X &= \\set{x_i | \\exists s \\in S: X(s) = x_i} \\\\
      R_X &= \\set{x_0, x_1, x_2, \\dots} \\\\
      \\end{align}
      $$<\/p>\n\n\n\n

      <\/span>Expectation<\/span><\/h2>\n\n\n\n

      https:\/\/www.probabilitycourse.com\/chapter3\/3_2_2_expectation.php<\/a><\/p>\n\n\n\n

      $$
      \\begin{align}
      E[X] &= \\sum_{x_k \\in R_X} x_k P_X(x_k) \\\\
      E[X] &= \\mu_X \\\\
      \\end{align}
      $$<\/p>\n\n\n\n

      <\/span>Linearity of Expectation<\/span><\/h3>\n\n\n\n

      $$
      \\begin{align}
      E[aX + b] &= \\sum_{x_k \\in R_X} (a x_k + b) P_X(x_k) &&& a, b \\in \\mathbb{R} \\\\
      E[aX + b] &= \\sum_{x_k \\in R_X} a x_k P_X(x_k) + \\sum_{x_k \\in R_X} b P_X(x_k) &&& a, b \\in \\mathbb{R} \\\\
      E[aX + b] &= a \\sum_{x_k \\in R_X} x_k P_X(x_k) + b \\sum_{x_k \\in R_X} P_X(x_k) &&& a, b \\in \\mathbb{R} \\\\
      E[aX + b] &= a E[X] + b &&& a, b \\in \\mathbb{R} \\\\
      \\end{align}
      $$<\/p>\n\n\n\n

      <\/span>Variance<\/span><\/h2>\n\n\n\n

      $$
      \\begin{align}
      Var[X] &= E[ (X – \\mu_X)^2] \\\\
      Var[X] &= \\sum_{x_k \\in R_X} (x_k – \\mu_X)^2 P_X(x_k) \\\\
      \\end{align}
      $$<\/p>\n\n\n\n

      Alternatively, it is common to express Variance as<\/p>\n\n\n\n

      $$
      \\begin{align}
      Var[X] &= \\sum_{x_k \\in R_X} (x_k – \\mu_X) (x_k – \\mu_X) P_X(x_k) \\\\

      Var[X] &= \\sum_{x_k \\in R_X} (x_k^2 – 2 \\mu_X x_k + \\mu_X^2) P_X(x_k) \\\\

      Var[X] &= \\sum_{x_k \\in R_X} x_k^2 P_X(x_k) – 2 \\mu_X \\sum_{x_k \\in R_X} x_k P_X(x_k) + \\mu_X^2 \\sum_{x_k \\in R_X} P_X(x_k) \\\\

      Var[X] &= \\sum_{x_k \\in R_X} x_k^2 P_X(x_k) – 2 \\mu_X \\mu_X + \\mu_X^2 \\\\

      Var[X] &= \\sum_{x_k \\in R_X} x_k^2 P_X(x_k) – 2 \\mu_X^2 + \\mu_X^2 \\\\

      Var[X] &= \\sum_{x_k \\in R_X} x_k^2 P_X(x_k) – \\mu_X^2 \\\\

      Var[X] &= E[X^2] – \\mu_X^2 \\\\

      \\end{align}
      $$<\/p>\n\n\n\n

      <\/span>Nonlinearity of Variance<\/span><\/h3>\n\n\n\n

      $$
      \\begin{align}
      Var[aX + b] &= \\sum_{x_k \\in R_X} \\{ a x_k + b – E[aX + b] \\}^2 P_X(x_k) &&&a, b \\in \\mathbb{R} \\\\
      Var[aX + b] &= \\sum_{x_k \\in R_X} \\{ a x_k + b – (a E[X] + b) \\}^2 P_X(x_k) \\\\
      Var[aX + b] &= \\sum_{x_k \\in R_X} \\{ a (x_k – E[X]) \\}^2 P_X(x_k) \\\\
      Var[aX + b] &= a^2 \\sum_{x_k \\in R_X} (x_k – E[X])^2 P_X(x_k) \\\\
      Var[aX + b] &= a^2 Var[X] \\\\
      \\end{align}
      $$<\/p>\n\n\n\n

      <\/p>\n\n\n\n

      <\/p>\n\n\n\n

      <\/p>\n\n\n\n

      <\/p>\n\n\n\n

      <\/p>\n","protected":false},"excerpt":{"rendered":"

      General An outcome $s$ is in sample space $S$ $$\\begin{align}s &\\in S \\\\S &= \\set{s_0, s_1, s_2, \\dots}\\end{align}$$ An event $E$ is a subset of samples in the sample space $$\\begin{align}E \\subset S\\end{align}$$ A Probability is a mapping of an event to a number that indicates how likely that event is to occur. Some Axioms: […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":106,"menu_order":3,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-523","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/523","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=523"}],"version-history":[{"count":45,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/523\/revisions"}],"predecessor-version":[{"id":595,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/523\/revisions\/595"}],"up":[{"embeddable":true,"href":"https:\/\/neilfoxman.com\/index.php?rest_route=\/wp\/v2\/pages\/106"}],"wp:attachment":[{"href":"https:\/\/neilfoxman.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=523"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}