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+#set page(paper: "a4", flipped: true, margin: 0.5cm, columns: 4)
+#set text(size: 9pt)
+#set list(spacing: 1.2em)
+
+- *Mututally Exclusive* - $A inter B = emptyset$
+- *Union* - $A union B = { x : x in A or x in B }$
+- *Intersection* - $A inter B = { x : x in A and x in B }$
+- *Complement* - $A' = { x : x in S and x in.not A }$
+- $(A inter B)' = (A' union B')$
+- *Multiplication* - R experiments performed sequentially. Then $n_i dot ... dot n_r$ possible outcomes for $r$ experiments
+- *Addition* - $e$ can be performed $k$ ways, and $k$ ways do not overlap : total ways: $n_1 + ... + n_k$
+- *Permutation* - Arrangement of $r$ objects out of $n$, _ordered_. $P^n_r = n!/(n-r)!, P^n_n = n!$
+- *Combination* - Selection of $r$ objects out of $n$, _unordered_ $vec(n, r) = n!/(r!(n-r)!), vec(n, r) times P^r_r = P^n_r$
+
+== Probability
+- Axioms:
+  + $0 <=P(A) <= 1$
+  + $P(S) = 1$
+- Propositions:
+  + $P(emptyset) = 0$
+  + $A_1 ... A_n$ are mutually exclusive,$P(A_1 union ... union A_n) = P(A_1) + ... + P(A_n)$
+  + $P(A') = 1-P(A)$
+  + $P(A) = P(A inter B) + P(A inter B')$
+  + $P(A union B) = P(A) + P(B) - P(A inter B)$
+  + If $A subset B, P(A) <= P(B)$
+
+== Conditional Probability
+- $P(B|A)$ is probability of $B$ given that $A$ has occured
+- $P(B|A) = P(A inter B) / P(A)$
+- $P(A inter B) = P(B|A)P(A)$
+- $P(A|B) = (P(A)P(B|A)) / P(B)$
+- $P(A inter B inter C) = P(A)P(B|A)P(C|B inter A)$
+- *Independent* - $P(A inter B) = P(A)P(B), A perp B$
+  - If $P(A) != 0, A perp B arrow.l.r P(B|A) = P(B)$ (Knowledge of $A$ does not change $B$)
+- *Independence vs Mutually exclusive* -
+  - $P(A) > 0 and P(B) > 0, A perp B arrow.double.r "not mutually exclusive"$
+- Partition - $A_i...A_n$ is mutually exclusive and $union.big^n_i=1 A_i = S, A_i...A_n$ is partition of S
+  - $P(B) = sum^n_(i=1) P(B inter A_i) = sum^n_(i=1) P(A_i)P(B|A_i)$
+  - $n = 2, P(B) = P(A)P(B|A) + P(A')P(B|A')$
+- *Bayes Theorem* - $P(A_k|B) = (P(A_k)P(B|A_k)) / (sum^n_(i=1)P(A_i)P(B|A_i))$
+- $n = 2, P(A|B) = (P(A)P(B|A)) / (P(A)P(B|A) + P(A')P(B|A'))$
+
+== Random Variables
+- Notations:
+  - ${X = x} = {s in S : X(s) = x) in S$
+  - ${X in A} = {s in S : X(s) in A) in S$
+== Probability Distributions
+- PMF(_Discrete_) of $X - f(x) = cases(P(X=x) "if" x in R_X, 0 "otherwise")$
+- Properties (*must* satisfy)
+  + $f(x_i) >= 0, x_i in R_X$
+  + $f(x_i) = 0, x_i in.not R_X$
+  + $sum^infinity_(i=1)f(x_i) = 1$
+- PDF(_Continuous_) of $X$ is function that satisfies the following
+  + $f(x) >= 0, x in R_X "and" f(x) = 0, x in.not R_X$
+  + $integral_R_X f(x) dif x = 1$
+  + $a <= b, P(a <= X <= b) = integral^b_a f(x) dif x$
+  - To validate, check (1) and (2)
+
+- CDF (Discrete) - $F(X) = P(X <=x)$
+  - $P(a<=X<=b) = P(X<=b) - P(X<a) = F(b) - F(a-), a-$ (is largest value in $R_X$ smaller than $a$)
+- CDF(Continuous) - $F(X) = integral^x_(-infinity)f(t)dif t$
+  - $f(x) = dif/(dif t) F(x)$
+  - $P(a<=X<=b) = F(b) - F(a)$
+  - $F(x)$ is non-decreasing.
+  - PDF/PMF and CDF have 1 to 1 correspondence.
+  - Ranges of $F(x)$ and $f(x)$ satisfy:
+    - $0 <= F(X) <= 1$
+    - for discrete: $0 <= f(X) <= 1$
+    - for continuous: $f(x) >= 0$, not necessary $f(x) <= 1$
+
+== Expectation
+- Expectation(Discrete): $ E(X) = mu_X = sum_(x_i in R_X) x_i f(x_i) $
+- Expectation(Continuous): $ E(X) = mu_X = integral^(infinity)_(-infinity)x_i f(x_i) $
+- *Properties*
+  + $E(a X + b) = a E(X) + b$
+  + $E(X + Y) = E(X) + E(Y)$
+  + Let $g(dot)$ be arbitrary function.
+    - $ E[g(X)] = sum g(x)f(x) \ "or"\ E[g(X)] = integral_R_X g(x)f(x)  $
+    - example - $E(X^2) = sum x^2f(x)$
+- Variance - $ sigma^2_x = V(X) = E(X - mu)^2 = E(X^2) - E(X)^2 $
+  - Discrete - $V(X) = sum (x-mu_x)^2 f(x)$
+  - Continuous - $V(X) = integral^(infinity)_(-infinity) (x-mu_x)^2 f(x)$
+  - *Properties*
+    + $V(a X + b) = a^2V(X)$
+    + $V(X) = E(X^2) - E(X)^2$
+    + Standard Deviation = $sigma_x = sqrt(V(X))$
+
+== Joint Probability Function
+- Discrete $ f_(X, Y)(x,y) = P(X = x, Y = y) $
+- *Properties*
+  + $f(X,Y)(x, y) >= 0, (x, y) in R_(X,Y)$
+  + $f(X,Y)(x, y) = 0, (x, y) in.not R_(X,Y)$
+  + $sum^infinity_(i=1)sum^infinity_(j=1)(x_i,y_i) = 1$
+- Continuous $ P((X, Y) in D) = integral.double_((x, y) in D) f(x,y) dif y dif x $
+  - $P(a<=X<=b, c<=Y<=d) = integral^b_a integral^d_c f(x, y) dif y dif x$
+- *Properties*
+  + $f_(X,Y)(x, y) >= 0$, for any $(x,y) in R_(X,Y)$
+  + $f_(X,Y)(x, y) = 0$, for any $(x,y) in.not R_(X,Y)$
+  + $integral^infinity_(-infinity)integral^infinity_(-infinity)f_(X,Y)(x, y) dif x dif y= 1$
+
+=== Marginal Probability Distribution
+  - Discrete - $f_X (x) = sum_y f_(X,Y)(x,y)$
+  - Continuous - $f_X (x) =integral^infinity_(-infinity) f_(X,Y)(x,y) dif y$
+
+- Conditional Distribution - $ f_(Y|X) (y|x) = (f_(X,Y)(x,y)) / (f_X (x)) $
+  - If $f_X (x) > 0, f_(X,Y)(x,y) = f_X (x) f_(Y|X) (y|x)$
+  - $P(Y <= y | X = x) = integral^y_(-infinity) f_(Y|X)(y|x) dif y$
+  - $E(Y | X = x) = integral^infinity_(-infinity) y f_(Y|X)(y|x) dif y$