From f1733a2433a4780322a7d74ce9cbe36deb9375c7 Mon Sep 17 00:00:00 2001 From: Holden Rohrer Date: Wed, 6 Oct 2021 15:18:51 -0400 Subject: notes and homeworks for math --- zhilova/08_jensen | 37 +++++++++++++++++++++++++++++++++++++ 1 file changed, 37 insertions(+) (limited to 'zhilova/08_jensen') diff --git a/zhilova/08_jensen b/zhilova/08_jensen index 20a8158..efe8bdb 100644 --- a/zhilova/08_jensen +++ b/zhilova/08_jensen @@ -21,3 +21,40 @@ f is strictly convex <=> f' is strictly increasing on (a,b) (2) If f is twice differentiable on (a,b) f is convex <=> f'' \geq 0 on (a,b) f is strictly convex <=> f'' > 0 on (a,b) + +Transformations of an r.v. + +Where X is an r.v. with pdf f_X(x), cdf F_X(x). + +Y := g(X). f_Y(y) = ? + +(Case 1) g is differentiable and invertible on D_X (range of X). + f_Y(y) = f_X(g^{-1}(y)) * |d/dy g^{-1}(y)| + Also: + if monotonically increasing, F_Y(y) = F_X(g^{-1}(y)) + if monotonically decreasing, F_Y(y) = 1 - F_X(g^{-1}(y)) + +(Case 2) g is piecewise bijective. +g is bijective on D_j where D_X = \cup_{j=1}^k D_j, with +D_i \cap D_j = \empty if i =/= j. (i.e. D_1...D_k is a partition of D_X) + +Then apply (1) through a sum. + f_y(y) = \sum f_X(g_j^{-1}(y)) * |d/dy g_j^{-1}(y)| * Indicator(y in range of g_j) + +F_Y(y) = P(g(X) \leq y) = P(\sum_{j=1}^k g_j(X) \leq y) += \int_R f_X(x) * indicator(x : g(x) \leq y) dx += \sum_{j=1}^k \int_R f_X(x) * Indicator{x : g_j(x) \leq y} dx +If g is monotonic increasing, the indicator is equivalent to x \leq +g_j^{-1}(y) + +This gives rise to several other transformations. Ex: scale +transformation ( g(x) = cx ), scale-position transformation ( g(x) = +cx+d ). + +Def: Symmetric distribution is when f_X(x) = f_X(-x). + +If X is a symmetric distribution and E|X| < \infty, EX = 0. + +EX = \int_{-\infty}^\infty xf_X(x) dx + = \int_0^\infty xf_X(x) + (-x)f_X(x) dx + = 0, by symmetry and some rearrangement of the integral. -- cgit