Transcript Continuous random variables

MATH 3033 based on
Dekking et al. A Modern Introduction to Probability and Statistics. 2007
Slides by Matthew Frankel
Instructor Longin Jan Latecki
Chapter 5: Continuous Random Variables
Probability Density Function of X
A random variable X is continuous if for some function ƒ: R  R and for any numbers a
and b with a ≤ b,
P(a ≤ X ≤ b) = ∫ab ƒ(x) dx
The function ƒ has to satisfy ƒ(x) ≥ 0 for all x and ∫-∞∞ ƒ(x) dx = 1.
Whereas Discrete Random Variables {px(a) =P(X=a)} map: Ω -- (X) -> R -- (px) -> [0,1]
Continuous Random Variables {P(a ≤ X ≤ b)=∫ab ƒ(x) dx} map: Ω -- (X) -> R -- (ƒ) -> R
To approximate the probability density function at a point a, one must find an ε
that is added and subtracted from a and then the area of the box under the curve
is obtained by the following: (2ε) * ƒ(a). As ε approaches zero the area under the
curve becomes more precise until one obtains an ε of zero where the area under
the curve is that of a width-less box. This is shown through the following
equation.
P(a – ε ≤ X ≤ a +ε) = ∫a-εa+ε ƒ(x) dx ≈ 2ε*ƒ(a)
A few asides:
DISCRETE  NO DENSITY
CONTINUOUS  NO MASS
BOTH  CUMULATIVE DISTRIBUTION Ƒ(a) = P(X ≤ a)
P(a < X ≤ b) = P(X ≤ b) – P(a ≤ X) = Ƒ(b) – Ƒ(a)
Ƒ(b) = ∫-∞b ƒ(x) dx and ƒ(x) = (d/dx) Ƒ(x)
*How the Distribution Function relates to the Density Function*
Uniform Distribution U(α,β)
•
A continuous random variable has a uniform distribution on the interval [α,β] if its
probability density function ƒ is given by ƒ(x) = 0 if x is not in [α,β] and,
ƒ(x) = 1/(β-α) for α ≤ x ≤ β
This simply means that for any x in the interval of alpha to beta has the same
probability and anything not in the interval is zero as shown in the figure below.
Exponential Distribution Exp(λ)
A continuous random variable has an exponential distribution
with parameter λ if its probability density function ƒ is given
by ƒ(x) = λe-λx for x ≥ 0
The Distribution function ƒ of an Exp(λ) distribution is given by
Ƒ(a) = 1 – e-λa for a ≥ 0
P(X > s + t | x > s = P(x > s + t)/P(x>s) = (e-λ(s+t))/(e-λs) =e-λt= P(X > t)
This simply means that s becomes the origin where t increases
therefore making s always less than t and the equation proven
true.
Pareto Distribution Par(α)
Simply used for estimating real-life situations such as class differences, city sizes,
earthquake rupture areas, insurance claims, and sizes of commercial companies.
A continuous random variable has a Pareto distribution with parameter α > 0 if its
probability density function ƒ is given by ƒ(x) = 0 if x > 1 and
f (x) 


x  1 for x ≥ 1
Normal Distribution N(μ,σ2)
Normal Distribution (Gaussian Distribution) with parameters μ and σ2 > 0 if its
probability density function ƒ is given by
1 x 
 (
1
f (x) 
e 2
 2

)2
for -∞ < x < ∞
*Where μ = mean and σ2 = standard deviation*
Distribution function is given by:

1 x 2
)

a
1  2(
F ( x)  
e
  2
dx
for -∞ < a < ∞
However, since ƒ does not have an antiderivative there is no explicit expression for Ƒ.
Therefore standard normal distribution where N(0,1) is given as follows, and the
distribution function is obtained similarly denoted by capital phi.
1
1  2 x2
 (x) 
e
2
for -∞ < x < ∞
Normal Distribution
Quantiles
Portions of the whole which increase from left to right, meaning
the 0th percentile is on the left hand side and the 100th
percentile is on the right side.
Let X be a continuous random variable and let p be a number
between 0 and 1. The pth quantile or 100pth percentile of
the distribution of X is the smallest number qp such that
Ƒ(qp) = P(X ≤ qp) = p
The median is the 50th percentile