Download Continuous Probability Distributions

Continuous Probability
Distributions
Previously we have been discussing the
probabilities associated with discrete
random variables – where a rv can only
assume a select number of values.
Now, we’ll extend this concept to continuous
random variables – where a rv can assume
any value within a specified range.
Definition
Continuous Random Variable: a rv with
a set of possible values in an entire
interval of numbers, A and B, where
A ≤ B.
Examples: height of tree, time waiting for
service, age, etc.
PDF of Continuous Rv
The probability distribution of probability
density function (pdf) of X is a function f(x)
such that for any two numbers a and b,
a ≤ b:
b
P ( a ≤ X ≤ b ) = ∫ f ( x ) dx
a
Conditions
f ( x) ≥ 0 , ∀ x
+∞
∫−∞ f ( x ) dx = area under the entire curve f ( x ) = 1
The probability that X takes on a value in
the interval [a,b] is the area under the
graph of the density function:
f(x)
a
b
x
Figure 1. Continuous Distribution
CDF for Continuous RV
The cumulative distribution function (cdf),
F(x), is defined for x as the area under the
curve to the left of x:
x
F ( X ) = P ( X ≤ x) = ∫
f ( y ) dy
−∞
Expected Value and Variance of
Continuous Distributions
The mean of a continuous random variable X
with a pdf f(x) is:
µx = E ( x ) = ∫
∞
−∞
xf ( x )dx
If X is a continuous rv with pdf f(x) and h(x) is
any function of X, then:
E ⎡⎣h ( x ) ⎤⎦ = ∫
∞
−∞
h ( x ) f ( x )dx
The variance of a continuous rv X with a pdf
f(x) and mean value µ is:
∞
2
2⎤
2
⎡
σx =V x =
x − µ f x dx = E X − µ
⎢⎣
⎥⎦
−∞
( ) ∫ (
) ( )
( ) − ⎡⎣E ( X )⎤⎦
2
=E X
2
(
)
The probability of X equally any number
c, P(X=c) = 0 and for any two numbers a
and b with a < b:
P ( a ≤ X ≤ b) = P ( a < X ≤ b) = P ( a ≤ X < b) = P ( a < X < b)
Uniform Distribution
A distribution that has constant probability. A
continuous rv is said to have uniform
distribution on the interval [A, B] if the pdf of
X is:
⎧ 1
A≤ x≤ B
⎪
f ( x; A, B ) = ⎨ B − A
⎪⎩ 0
otherwise
Uniform Example
Suppose every 10 minutes a bus arrives at
your stop. Due to the variation in the time
you leave your house, you don’t always get
to the bus stop at the same time.
Therefore, the time spent waiting for the bus,
X, is a continuous random variable.
What is the probability that you wait between 1
and 3 minutes?
Graphically
1
10
0
10
0
1
3
10
⎧ 1
⎪
f ( x;0,10 ) = ⎨10 − 0
⎪⎩ 0
0 ≤ x ≤ 10
otherwise
1
P (1 ≤ X ≤ 3 ) = ∫ f ( x )dx = ∫
dx
1
1 10
3
3
x =3
x
3
1
2
=
=
−
=
10 x =1 10 10 10
1
= = 0.20
5
Normal Distribution
The most well known distribution is the
normal distribution.
Also referred to as the Gaussian
distribution, after Carl Friedrich Gauss,
or the “bell curve”, after the shape of
the distribution:
PDF and CDF of Normal Distribution
The normal density is given by:
1
−( x − µ )
f ( x) =
e
2πσ
2
2σ 2
,
−∞ < x < ∞
where, π and e are the constants ~ 3.14
and 2.72; and µ and σ are the mean and
standard deviation, respectively
Characteristics of the
Normal Distribution
„
„
„
„
„
„
„
„
the distribution is symmetric about the mean
the mean = median = mode
large standard deviations result in flatter curves
smaller standard deviations result in taller
curves
µ ± σ ~68% of the distribution area
µ ± 2σ ~95% of the distribution area
µ ± 3σ ~99.7% of the distribution area
the mean and standard deviation completely
define the distribution: X ~ N(µ, σ)
Standard Normal ~N(0,1)
The standard normal distribution, is a
normal distribution with a mean = 0 and
standard deviation = 1:
1 −( z )
f ( z) =
e
2π
2
x − µ)
(
z=
σ
2
,
−∞ < z < ∞
Graphically
0.08
0.05
0.03
-3.0
-1.0 .0 1.0 2.0 3.0 4.0
Probability
0.10
CDF Normal
Cum Prob
0.9
0.7
0.5
0.3
0.1
-4 -3 -2 -1 0 1
Column 1
2
3
4
Of note:
Data generated from a N(µ, σ) will not
necessarily have a mean = µ and a
standard deviation = σ
Though the data was generated using a
normal distribution with mean = 0 and
standard deviation = 1, the estimated
mean = -0.02 and the estimated standard
deviation = 0.999.
Probability of X
The entire area under the curve of a standard
normal distribution is 1 unit.
Therefore, the relationship between the
normal distribution and the standard normal
distribution can be used to determine the
probability of z being ≤, < , ≥, >, or = to any
zO.
Other Distributions
There are numerous other continuous
distributions:
– Gamma
– Exponential
– Chi-Squared