notes and homework

1 files changed, 23 insertions, 0 deletions

diff --git a/zhilova/08_jensen b/zhilova/08_jensen
new file mode 100644
index 0000000..20a8158
--- /dev/null
+++ b/zhilova/08_jensen
@@ -0,0 +1,23 @@
+Definition: A fn. f : R -> R is called convex on an interval (a,b) if
+f(cx + dy) \leq cf(x) + df(y)
+\forall x, y \in (a, b)
+\forall c \in (0, 1), d = 1-c.
+Concave is -convex.
+
+Essentially stating that the function lies on or below a line segment
+connecting f(a) and f(b) [or above in the case of concave].
+
+Strictly convex: f(cx+dy) < cf(x) + df(y).
+
+    Jensen's Inequality
+X - r.v., E|X| < infty. E|f(x)| < infty.
+    f(E X) \leq E(f(x)).
+If f is strictly convex, \leq -> "less than" unless X is a constant r.v.
+
+Further theorems:
+(1) If f is differentiable on (a,b),
+f is convex <=> f' is nondecreasing on (a,b).
+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)