notes and homework

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+    Cumulative Distribution Function (CDF)
+Def: CDF of a r.v. X, taking values in R is
+F_X(x) = \Pr(X\leq x) = \Pr(X\in (-\infty, x] ) % to appease vim, ')'
+
+Th 1.5.1 (Properties of a CDF)
+0) 0 \leq F_X(x) \leq 1 \forall x \in R
+1) It is non-decreasing. x_1 \leq x_2 \in A, F_X(x_1) \leq F_X(x_2).
+2) F_X(x) -> 0 as x -> -\infty
+3) F_X(x) -> 1 as x -> +\infty
+4) F_X(x) is right-continuous.
+
+    Continuous R.V.
+Over an uncountable domain D like (0, 1), R.
+
+Let there be a CDF F_X(x) = P(X \leq x).
+
+Assume there exists f_X(x) := d/dx F_X(x), the probability density
+function.
+[discontinuities might be able to be resolved with a delta function]
+By the second fundamental theorem of calculus (?),
+F_X(x) = P(X \leq x) = \int_{-\infty}^\infty f_x(t) dt.
+
+In the discrete case, we have the pmf (probability mass function)
+where P_x(t) = P(X = t)
+
+P(a < X \leq b) for a < b = P_X(b) - P_X(a).
+
+Examples:
+- Uniform Distribution
+X ~ U[a, b]
+ = { 1/(b-a) for a \leq x \leq b
+   { 0 otherwise.
+
+- Exponential Distribution
+X ~ Exp(\lambda) \lambda > 0
+f_X(x) = { \lambda e^{-\lambda x}, x \geq 0
+         { 0 otherwise
+
+F_X(x) = { 1 - e^{-\lambda x}, x \geq 0
+         { 0 otherwise
+
+- Normal Distribution
+X ~ N(\mu, \sigma^2) \mu \in R, \sigma^2 > 0.
+\sigma = stdev. \sigma^2 = variance. \mu = mean/center.
+
+f_X(x) = 1/\sqrt{2\pi \sigma^2}  exp( - (x-\mu)^2 / {2\sigma^2} )
+
+F_X(x) = \int_{-\infty}^x f_X(x) dx