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 Continuous random variables

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.. 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