notes and homeworks for math

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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.