Probability density function. Continuous uniform random variable. Exponential
random variables. Cumulative distribution functions.
=============================
Continuous random variables
=============================
.. contents::
:local:
Probability density function
============================
.. role:: def
:class: def
:def:`Probability density function` (PDF) for continuous random variable
:math:`x` is function:
.. math::
PDF(a ≤ X ≤ b) = P(a ≤ X ≤ b) = ∫_{a, b} f_X(x) dx
f_X(x) ≥ 0
∫_{-∞, +∞} f_X(x) dx = 1
:math:`f_X(x)` funtion maps values :math:`x` from sample space to real numbers.
For continuous random variable:
.. math:: P(X = a) = 0
Expectation
===========
:def:`Expectation` of continuous random variable is:
.. math:: μ = E[X] = ∫_{-∞, +∞} x·f_X(x)·dx
Properties:
.. math::
E[X + Y] = E[X] + E[Y]
E[a·X] = a·E[X]
E[a·X + b] = a·E[X] + b
Variance
========
:def:`Variance` of continuous random variable is:
.. math:: var[X] = ∫_{-∞, +∞} (x-μ)²·f_X(x)·dx
Properties:
.. math::
var[a·X + b] = a²·var[X]
var[X] = E[X²] - E²[X]
Standard deviation
==================
:def:`Standard deviation` of continuous random variable is:
.. math:: σ_Χ = sqrt(var[X])
Continuous uniform random variable
==================================
:def:`Continuous uniform random variable` is :math:`f_X(x)` that is non-zero
only on :math:`[a, b]` with :math:`f_X(x) = `1/(b-a)`.
.. math::
E[unif(a, b)] = (b+a)/2
var[unif(a, b)] = (b-a)²/12
σ = (b-a)/sqrt(12)
Proofs:
.. math::
E[unif(a, b)] = ∫_{a, b} x·1/(b-a)·dx = x²/2/(b-a) |_{a, b} = (b²-a²)/(b-a)/2 = (b+a)/2
E[unif²(a, b)] = ∫_{a, b} x²·1/(b-a)·dx = x³/3/(b-a) |_{a, b} = (b³-a³)/(b-a)/3 = (b²+b·a+a²)/3
var[unif(a, b)] = E[unif²(a, b)] - E²[unif(a, b)] = (b²+b·a+a²)/3 - (b+a)²/4 = (b-a)²/12
.. note::
In maxima::
(%i4) factor((b^2+b*a+a^2)/3 - (a+b)^2/4);
2
(b - a)
--------
12
Exponential random variables
============================
:def:`Exponential random variables` with parameter :math:`λ` is:
.. math:: f_X(x) = λ·exp(-λ·x)
for :math:`x ≥ 0`, and zero otherwise.
Properties:
.. math::
E[exp(λ)] = 1/λ
var[exp(λ)] = 1/λ²
Proof:
.. math::
∫_{-∞, +∞} f_X(x)·dx = ∫_{0, +∞} λ·exp(-λ·x)·dx = -exp(-λ·x) |_{0, +∞} = 1
E[exp(λ)] = ∫_{0, +∞} x·λ·exp(-λ·x)·dx = 1/λ
E[exp²(λ)] = ∫_{0, +∞} x²·λ·exp(-λ·x)·dx = 1/λ²
.. note::
From maxima::
(%i15) assume(lambda>0);
(%o15) [lambda > 0]
(%i16) integrate(lambda*%e^(-lambda*x),x,0,inf);
(%o16) 1
(%i17) integrate(x*lambda*%e^(-lambda*x),x,0,inf);
1
(%o17) ------
lambda
(%i18) integrate(x^2*lambda*%e^(-lambda*x),x,0,inf);
2
(%o18) -------
2
lambda
Cumulative distribution functions
=================================
:def:`Cumulative distribution functions` of random variable :math:`X` is:
.. math:: F_X(x) = P(X ≤ x) = ∫_{-∞, x} f_X(t)·dt