Download Important parametric families of univariate distributions

Unit E
Important parametric families
univariate distributions
of
In this Unit we review some of the most important parametric
families of univariate distributions.
Discrete distributions
A r.v. X is discrete if it takes values on a finite or countable
set ⌦. We denote the probability function of a discrete r.v. by
f (x), for x 2 ⌦ (f (x) = 0 elsewhere) and the corresponding
distribution function by F (x).
The Bernoulli distribution
• A Bernoulli r.v. X with parameter p 2 [0, 1] takes values
in ⌦ = {0, 1} and has p.f.
f (x) = px(1
p)1 x,
for x 2 {0, 1}.
• We will use the notation X ⇠ Ber(p).
• Its expected value and variance are
E(X) = p,
V ar(X) = p(1
p)
The Binomial distribution
• A Binomial r.v. X with parameters n = 1, 2, . . ., and
p 2 [0, 1] takes values in ⌦ = {0, 1, 2, . . . , n} and has
p.f.
✓ ◆
n x
f (x) =
p (1 p)n x,
x
for x 2 {0, 1, . . . , n}.
Important parametric families of . . .
174
• We will use the notation X ⇠ Bin(n, p).
• Its expected value and variance are
E(X) = np,
V ar(X) = np(1
p).
• Bin(n, p) is used as a model for the number of successes
in n independent trials when at each trial the probability
of observing a success is p.
• If X1, X2, . . . , Xn are independent Bernoulli r.v.’s with
parameter p,
X1 + X2 + . . . + Xn ⇠ Bin(n, p).
The Geometric distribution
• A Geometric r.v. X with parameter p 2 [0, 1] takes values
in ⌦ = {1, 2, . . .} and has p.f.
f (x) = p(1
p)x 1,
for x 2 {1, 2, . . .}.
• We will use the notation X ⇠ Geo(p).
• Its expected value and variance are
1
E(X) = ,
p
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Unit E:
V ar(X) =
p
1
p2
.
• Geo(p) is used as a model for the number of independent
trials till failure, when at each trial the probability of
observing a failure is p.
The Poisson distribution
• A Poisson r.v. X with parameter
⌦ = {0, 1, . . .} and has p.f.
f (x) =
> 0 takes values in
x
e
x!
,
for x 2 {0, 1, . . .}.
• We will use the notation X ⇠ Pois( ).
• Its expected value and variance are
E(X) = ,
V ar(X) = .
• Pois( ) is used as a model for the number of events
occurring in a fixed period of time if the events occur
independently.
• If X1, . . . , Xn are independent r.v.’s and Xi ⇠ Pois( i),
then
n
X
X1 + X2 + . . . + Xn ⇠ Pois(
i ).
i=1
Important parametric families of . . .
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Continous distributions
A r.v. X is continous if it takes an uncountable number values,
i.e., it takes values in an interval of R. We denote the density
function of a continous r.v. by f (x) and the corresponding
distribution function by F (x).
The Uniform distribution
• A Uniform r.v. X takes values on an interval (a, b) ⇢ R
and has d.f.
(
1
, for a  x  b,
b
a
f (x) =
0,
otherwise.
• We will use the notation X ⇠ U(a, b).
• Its expected value and variance are
E(X) =
V ar(X) =
b+a
,
2
(b
a)2
.
12
The Normal distribution
177
• A Normal (or Gaussian) r.v. X with parameters µ 2 R
and 2 > 0 takes values in R and has d.f.
⇢
1
1
2
f (x) = p
exp
(x
µ)
,
2
2
2
2⇡
Unit E:
for x 2 R.
• We will use the notation X ⇠ N (µ,
2
).
• Its expected value and variance are
E(X) = µ,
V ar(X) =
2
.
• Its d.f. has a “bell” shape symmetric around µ and with
dispersion controlled by 2.
• If X ⇠ N (µ,
2
), for any real constants a and b,
a + bX ⇠ N (a + bµ, b2 2).
• The distribution function of N (µ,
F (x) =
Z
2
)
x
f (y)dy
1
has no closed form. However, the distribution function of
N (0, 1) (the standard Normal distribution), denoted
by (x), is tabulated and F (x) can be derived from
F (x) =
✓
x
µ
◆
by the previous property.
Important parametric families of . . .
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• If X1, X2, . . . , Xn are independent and Xi ⇠ N (µi,
then
X1 + X2 + . . . + Xn ⇠ N (
n
X
µi ,
i=1
n
X
2
i ),
2
i ).
i=1
• The Normal distribution has a key role in Probability
Theory by virtue of the Central Limit Theorem which,
broadly speaking, states that the sum of r.v.’s can be
approximated by a Gaussian r.v. with appropriate mean
and variance.
The Exponential distribution
• An Exponential r.v. X with parameter > 0 takes values
in R+ and has d.f.
(
e x, for x > 0,
f (x) =
0,
otherwise.
• We will use the notation X ⇠ Exp( ).
• Its distribution function is
Z x
F (x) =
e
y
dy = 1
e
x
.
0
• Its expected value and variance are
E(X) = 1/ ,
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Unit E:
V ar(X) = 1/ 2.
• The Exponential r.v. is used as a model for the
lifetime of organisms or technical devices not subject to
deterioration and for times between events when events
occur independently and at a constant rate.
The Gamma distribution
• A Gamma r.v. X with parameters > 0 and ↵ > 0
takes values in R+ and has d.f.
( ↵ ↵ 1 x
x
e
, for x > 0,
(↵)
f (x) =
0,
otherwise,
where (↵) is the Gamma function
(↵) =
Z
1
x↵ 1e xdx.
0
• We will use the notation X ⇠ Gamma( , ↵).
• Its mean and variance are
E(X) = ↵/ ,
V ar(X) = ↵/ 2.
• The Gamma function must in general be computed by
numerical integration. However, if ↵ is an integer, (↵) =
(↵ 1)!.
Important parametric families of . . .
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• The distribution function F (x) of a Gamma( , ↵) must
in general be computed by numerical integration.
• For ↵ = 1, the Gamma( , ↵) distribution becomes the
Exp( ) distribution.
• If X1, X2, . . . , Xn are independent r.v.’s and Xi ⇠
Gamma( , ↵i), then
X1 + X2 + . . . + Xn ⇠ Gamma( ,
n
X
↵i).
i=1
In particular, if X1, X2, . . . , Xn are independent r.v.’s and
Xi ⇠ Exp( ), then
X1 + X2 + . . . + Xn ⇠ Gamma( , n).
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Unit E:
Important parametric families of . . .
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