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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
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