STAT355 - Probability & Statistics
Chapter 4:
Continuous Random Variables and Probability
Distributions
Instructor: Kofi Placid Adragni
Fall 2011
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
Fall
Chapter
2011 4: Continuo
1 / 21
Chap 4 - Continuous Random Variables and Probability
Distributions
1
4.1 Probability Density Function
2
4.2 The Cumulative Distribution Function and Expected Values
3
4.3 The Normal Distribution
4
4.4 The Exponential and Gamma Distributions
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
Fall
Chapter
2011 4: Continuo
2 / 21
Probability Density Function
A discrete random variable (rv) is one whose possible values either
constitute a finite set or else can be listed in an infinite sequence (a list in
which there is a first element, a second element, etc.).
A random variable whose set of possible values is an entire interval of
numbers is not discrete.
Definition
Let X be a continuous rv. Then a probability distribution or probability
density function (pdf) of X is a function f (x) such that for any two
numbers a and b with a ≤ b,
Z
P(a ≤ X ≤ b) =
b
f (t)dt
a
Remark: The graph of f (x) is often referred to as the density curve.
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
Fall
Chapter
2011 4: Continuo
3 / 21
Probability Density Function
I For f (x) to be a legitimate pdf, it must satisfy the following two
conditions:
1
f (x) ≥ 0 for all x
2
R∞
−∞ f (t)dt
=1
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
Fall
Chapter
2011 4: Continuo
4 / 21
Exercise
(4.1) 2
Suppose the reaction temperature X (in ◦ C ) in a certain chemical process
has a uniform distribution with a = −5 and b = 5.
1 Compute P(X < 0).
2 Compute P(−2.5 < X < 2.5).
3 Compute P(−2 ≤ X ≤ 3).
4 For k satisfying −5 < k < k + 4 < 5, compute P(k < X < k + 4).
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
Fall
Chapter
2011 4: Continuo
5 / 21
The Cumulative Distribution Function and Expected Values
Definition
The cumulative distribution function F (x) for a continuous rv X is defined
for every number x by
Z x
F (x) = P(X ≤ x) =
f (t)dt
−∞
Proposition
Let X be a continuous rv with pdf f (x) and cdf F (x). Then for any
number a,
P(X > a) = 1 − F (a)
and for any two numbers a and b with a < b,
P(a ≤ X ≤ b) = F (b) − F (a)
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
Fall
Chapter
2011 4: Continuo
6 / 21
Example
Let X , the thickness of a certain metal sheet, have a uniform distribution
on [a, b]. Recall that the pdf of X is
1
if a ≤ x ≤ b
b−a
f (x) =
0
otherwise
Find the cdf and graph it.
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
Fall
Chapter
2011 4: Continuo
7 / 21
More on CDF
Proposition
If X is a continuous rv with pdf f (x) and cdf F (x), then at every x at
0
0
which the derivative F (x) exists, F (x) = f (x).
Definition
Let p be a number between 0 and 1. The (100p)th percentile of the
distribution of a continuous rv X , denoted by η(p), is defined by
Z
η(p)
p = F (η(p)) =
F (y )dy
−∞
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
Fall
Chapter
2011 4: Continuo
8 / 21
Example
The distribution of the amount of gravel (in tons) sold by a particular
construction supply company in a given week is a continuous rv X with pdf
3
2
if 0 ≤ x ≤ 1
2 (1 − x )
f (x) =
0
otherwise .
I The cdf of sales for any x between 0 and 1 is
x3
3
F (x) = (x − )
2
3
I The (100p)th percentile of this distribution satisfies the equation
η(p)3
3
)
p = F (η(p)) = (η(p) −
2
3
I To obtain the median (50th percentile, p = 0.50), solve the equation
η 3 − 3η + 1 = 0
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
Fall
Chapter
2011 4: Continuo
9 / 21
Expected Values
Definition
The expected or mean value of a continuous rv X with pdf f (x) is
Z ∞
µX = E (X ) =
t f (t)dt.
−∞
Proposition
If X is a continuous rv with pdf f (x) and h(X ) is any function of X , then
Z ∞
E [h(X )] = µh(X ) =
h(x)f (x)dx
−∞
Example: Let X be a rv with an uniform distribution on [a, b]. Find the
expected value of X .
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
FallChapter
2011 4: 10
Continuo
/ 21
Variance
Definition
The variance of a continuous random variable X with pdf f (x) and mean
value σX2 is
Z ∞
2
(x − µ)2 f (x)dx = E [(X − µ)2 ]
σX = V (X ) =
−∞
The standard deviation (SD) of X is σX =
p
V (X )
Proposition
V (X ) = E (X 2 ) − [E (X )]2
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
FallChapter
2011 4: 11
Continuo
/ 21
Exercise
(4.2) 11
Let X denote the amount of time a book on two-hour reserve is actually
checked out, and suppose the cdf is
if x < 0
0
x2
F (x) =
if
0
≤x <2
4
1
if 2 ≤ x
Use the cdf to obtain the following:
1 P(X ≤ 1)
2 P(0.5 ≤ X ≤ 1)
3 P(X > 1.5)
4 The median checkout duration µ̃ [ solve 0.5 = F (µ̃)]
0
5 F (x) to obtain the density function f (x)
6 E (X )
7 V (X ) and σ
X
8 If the borrower is charged an amount h(X ) = X 2 when checkout
duration is X , compute the expected charge E [h(X )].
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
FallChapter
2011 4: 12
Continuo
/ 21
The Normal Distribution
Definition
A continuous rv X is said to have a normal distribution with parameters µ
and σ 2 where −∞ < µ < infty and σ > 0, if the pdf of X is
f (x; µ, σ 2 ) = √
1
1
exp{− 2 (x − µ)2 }
2σ
2πσ
−∞<x <∞
Notation: X is N(µ, σ 2 ) or X ∼ N(µ, σ 2 ).
Remarks:
f (x; µ, σ 2 ) ≥ 0
R∞
2
−∞ f (x; µ, σ )dx = 1
It can be shown that E (X ) = µ and V (X ) = σ 2
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
FallChapter
2011 4: 13
Continuo
/ 21
Remarks on the Normal Curve
Each density curve is symmetric about µ and bell-shaped, so the
center of the bell (point of symmetry) is both the mean of the
distribution and the median.
The value of σ is the distance from µ to the inflection points of the
curve (the points at which the curve changes from turning downward
to turning upward).
Large values of σ yield graphs that are quite spread out about µ,
whereas small values of σ yield graphs with a high peak above µ and
most of the area under the graph quite close to µ.
Thus a large σ implies that a value of X far from µ may well be
observed, whereas such a value is quite unlikely when σ is small.
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
FallChapter
2011 4: 14
Continuo
/ 21
The Standard Normal
Definition
The normal distribution with parameter values µ = 0 and σ = 1 is called
the standard normal distribution. A random variable having a standard
normal distribution is called a standard normal random variable and will be
denoted by Z . The pdf of Z is
1
1
f (z; µ, σ 2 ) = √ exp{− z 2 }
2
2π
−∞<z <∞
The cumulative distribution function of Z is often denoted by
Z z
1
1
√ exp{− t 2 }dt
Φ(z) = P(Z ≤ z) =
2
2π
−∞
There is no analytical expression of Φ(z). For specified values of z, Table
A-3 in Textbook provides the values of Φ(z).
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
FallChapter
2011 4: 15
Continuo
/ 21
Examples
Find:
P(Z ≤ 0)
P(Z < 1.5)
P(Z ≥ 1.5)
P(Z < −1.5)
P(Z ≤ 3)
P(Z > −2)
Remark: P(Z ≤ −z) = 1 − P(Z ≤ z)
Notation: zα will denote the value on the z axis for which α of the area
under the z curve lies to the right of zα .
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
FallChapter
2011 4: 16
Continuo
/ 21
Nonstandard Normal Distributions
Proposition
If X has a normal distribution with mean µ and standard deviation σ, then
Z=
X −µ
σ
has a standard normal distribution. Thus
P(X ≤ a) = Φ( a−µ
σ )
P(X ≥ b) = 1 − Φ( b−µ
σ )
P(a ≤ X ≤ b) = P( a−µ
σ ≤Z ≤
b−µ
σ )
a−µ
= Φ( b−µ
σ ) − Φ( σ )
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
FallChapter
2011 4: 17
Continuo
/ 21
The Exponential Distribution
Definition
X is said to have an exponential distribution with parameter λ (λ > 0) if
the pdf of X is
λe −λx
if x ≥ 0
f (x; λ) =
0
otherwise .
Proposition
If X is a rv with an Exponential(λ) distribution, then its mean and
variance are
1
1
µ = and σ 2 = 2
λ
λ
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
FallChapter
2011 4: 18
Continuo
/ 21
The Gamma Function
Definition
For α > 0, the gamma function Γ(α) is defined by
Z ∞
Γ(α) =
x α−1 e −x dx
0
Some properties of Γ(α)
For any α > 1, Γ(α) = (α − 1)Γ(α − 1)
For any positive integer, n, Γ(n) = (n − 1)!
√
Γ( 21 ) = π
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
FallChapter
2011 4: 19
Continuo
/ 21
The Gamma Distribution
Definition
A continuous random variable X is said to have a gamma distribution if
the pdf of X is
(
1
α−1 e −x/β
if x ≥ 0
β α Γ(α) x
f (x; α, β) =
0
otherwise .
where the parameters α and β satisfy α > 0, β > 0.
Proposition
Let X be a rv with a gamma(α, β) distribution. Then the mean and
variance of X are
µ = E (X ) = αβ;
σ 2 = V (X ) = αβ 2
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
FallChapter
2011 4: 20
Continuo
/ 21
More Distributions
The chi-square
The Weibull
Lognormal
Beta
...
STAT355 Instructor:
- ProbabilityKofi
& Statistics
Placid Adragni ()
FallChapter
2011 4: 21
Continuo
/ 21
0
