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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)
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.
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