MACHINE LEARNING TECHNIQUES - LASA

186
9.2.2 Probability Distributions, Probability Density Function
p(x) a continuous function is the probability density function or probability distribution function
(PDF) (sometimes also called probability distribution or simply density) of variable x. p(x) must be
Lebesgue integrable and :
f(x) is the probability distribution (also called the probability density function or the probability
distribution function) of variable x. f is a non-negative and normalized function:
px ( ) ≥0,
∫
∞
−∞
∀x
p( x) dx = 1
9.2.3 Distribution Function or Cumulative Distribution Function
* *
The distribution function D( x ) P( x x )
= ≤ also called the cumulative distribution function
(CDF) describes the probability that a variate X takes on value less than or equal to a number x. If
p(x) is the probability density function of x then:
d
P x p x dx p x P x
dx
( ) = ( ) ⇒ ( ) = ( )
∫
(8.20)
p(x) dx ~ probability of x to fall within an infinitesimal interval [x, x + dx].
The probability that the variable x takes a value in the subinterval [a,b] is equal to:
Pa ( ≤ x≤ b) =∫ f( xdx )
b
a
The expectation of x, E(x), is the mean µ of its distribution:
µ = Ex () = ∫ x⋅ f()
x⋅dx
(8.21)
If x follows a discrete distribution, then:
The variance
mean:
µ = Ex ( ) =∑ xf( x)
(8.22)
i
i
2
σ of a distribution measures the amount of spread of the distribution around its
2
(( ) ) ( ) ( ) 2
2 2
σ Var( x)
E x µ E x ⎡⎡E x ⎤⎤
i
= = − = − ⎣⎣ ⎦⎦
(8.23)
σ is the standard deviation of x.
© A.G.Billard 2004 – Last Update March 2011