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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 +- 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* - $P(A_k|B) = (P(A_k)P(B|A_k)) / (sum^n_(i=1)P(A_i)P(B|A_i))$ +- *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: +- 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 @@ -56,14 +56,14 @@ + $a <= b, P(a <= X <= b) = integral^b_a f(x) dif x$ - To validate, check (1) and (2) -- CDF (Discrete) - $F(X) = P(X <=x)$ +- CDF (Discrete): $F(X) = P(X <=x)$ - $P(a<=X<=b) = P(X<=b) - P(X= 0$, not necessary $f(x) <= 1$ @@ -74,12 +74,12 @@ - *Properties* + $E(a X + b) = a E(X) + b$ + $E(X + Y) = E(X) + E(Y)$ - + Let $g(dot)$ be arbitrary function. + + 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 - $V(X) = integral^(infinity)_(-infinity) (x-mu_x)^2 f(x)$ + - 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* + $V(a X + b) = a^2V(X)$ + $V(X) = E(X^2) - E(X)^2$ @@ -99,10 +99,68 @@ + $integral^infinity_(-infinity)integral^infinity_(-infinity)f_(X,Y)(x, y) dif x dif y= 1$ === Marginal Probability Distribution - - Discrete - $f_X (x) = sum_y f_(X,Y)(x,y)$ - - Continuous - $f_X (x) =integral^infinity_(-infinity) f_(X,Y)(x,y) dif y$ + - 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)) $ +- Conditional Distribution: $ f_(Y|X) (y|x) = (f_(X,Y)(x,y)) / (f_X (x)) $ - If $f_X (x) > 0, f_(X,Y)(x,y) = f_X (x) f_(Y|X) (y|x)$ - $P(Y <= y | X = x) = integral^y_(-infinity) f_(Y|X)(y|x) dif y$ - $E(Y | X = x) = integral^infinity_(-infinity) y f_(Y|X)(y|x) dif y$ +=== Independent Random Variable +- *Independent*: $ f_(X, Y)(x, y) = f_X(x) f_Y(y) $ +- *Properties* + + If $X, Y$ are independent random variables, $ P(X <= x; Y <= y) = P(X <= x) P(Y <= y) $ + + $g_1(X) "and" g_2(Y)$ are independent. (E.g. $X^2 "and" log(Y)$ are independent) + + if $F_X(x) > 0, "then" f_(Y|X)(y|x) = f_Y (y)$ + +#colbreak() +=== Expectation and Covariance +- *Expectation*: $ E(g(X, Y)) = sum_x sum_y g(x, y)f_(X, Y)(x, y) \ + E(g(X, Y)) = integral^(infinity)_(-infinity) integral^(infinity)_(-infinity) g(x, y)f_(X, Y)(x, y) dif y dif x $ + +#[ +#let cov = "cov" +- *Covariance*: $ cov(X, Y) = sum_x sum_y (x-mu_x)(y-mu_y)f_(X, Y)(x, y) $ +- *Properties* + + $cov(X, Y) = E(X Y) - E(X)E(Y)$ + - $E(X Y) = integral integral x y f(x, y) dif y dif x$ + + If $X$ and $Y$ are independent, $cov(X, Y) = 0$ + - $X perp Y => cov(X, Y) = 0$ + - $cov(X, Y) = 0 arrow.double.not X perp Y$ + - $E(X Y) = E(X)E(Y)$ + + $cov(a X + b, c Y + d) = a c dot cov(X, Y)$ + + $V(a X + b Y) = a^2V(X) + b^2V(Y) + 2 a b dot cov(X, Y)$ +] += Probability Distributions + +== Discrete Distributions +=== Discrete Uniform Distribution +If random variable $X$ assumes values $x_1, ...$ with _equal_ probability, then $X$ follows discrete uniform distribution. PMF of $X$ is $ f_X(x) = cases(1/k\, &x = x_1\,...\,x_k \ 0 & "otherwise") $ +$ E(X) = sum^k_(i=1)x_i f_X (x_i) = 1/k sum^k_(i=1)x_i $ +$ V(X) = E(X^2) - E(X)^2 = 1/k sum^k_(i=1)x_i^2 - mu^2_x $ + +- *Bernoulli Trial*: experiment with only 2 outcomes (1/0) +- *Bernoulli Random Variable*: X be no of sucess in Bernoulli trial. $X$ has only 2 values. $p$ is probability of success. PMF: $ f_X (x) &= P(X = x) = cases(p &"if" x = 1, 1-p &"if" x = 0) \ &= p^x (1-p)^(1-x), "for" x = 0,1 $ +- $X ~ "Bernoulli"(p)$ + - $q = 1-p$ + - $f_X (1) = p, f_X (0) = q$ + - $E(X) = p$ + - $V(X) = p q$ +- *Bernoulli Process* - repeated independent and identical bernoulli trials +- *Binomial Random Variable* - No of successes in $n$ trials of bernoulli process. + - P of $x$ successes in $n$ trials +- $X ~ "Bin"(n, p)$ + - $P(X = x) = vec(n,x)p^x (1-p)^(n-x)$ + - $E(X) = n p$ + - $V(X) = n p(1-p)$ +- *Negative Binomial Distribution* - No of trials needed for $k$ successes + - P of $x$ trials needed for $k$ successes +- $X ~ "NB"(k, p)$ + - $f_(x)(x) = P(X = x) = vec(x - 1, k - 1)p^(k)(1-p)^(x-k)$ + - $E(X) = k/p$ + - $V(X) = ((1-p)k) / p^2$ +- *Geometric Distribution*: No of trials needed until first success occurs. +- $X ~ "Geom"(p) = "NB"(1, p)$ + - $f_x (x) = P(X = x) = (1-p)^(x-1)p$ + - $E(X) = 1/p$ + - $V(X) = (1-p)/p^2$