Download Uniform, Normal and Gamma distributions

Uniform, Normal and Gamma
distributions
Uniform distribution
A random variable with unif[a,b] distribution has pdf
f (x; a, b) =
1
if x ∈ [a, b]
b−a
It can be checked that
E(X ) =
a+b
2
and
Var(X ) =
(b − a)2
.
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Normal distribution
The probability density function of a normal distribution (or
Gaussian distribution) is
f (x; µ, σ 2 ) = √
1
2πσ
exp{−
(x − µ)2
},
2σ 2
which will be denoted by N(µ, σ 2 ), where µ is a location
parameter and σ is a scale parameter. N(0, 1) is a standard
normal distribution. Typically, we use φ(x) to denote the pdf of
N(0, 1) and use Φ(x) to denote the CDF function of N(0, 1).
N(0,1)
N(1,1)
N(0,2)
0.2
0.1
0.0
Density f(x)
0.3
0.4
Normal density plots
−4
−2
0
x
2
4
Expectation and variance
I
If Z ∼ N(0, 1), then Y = µ + σZ ∼ N(µ, σ 2 ).
I
If Y ∼ N(µ, σ 2 ), then Z =
I
E(Z ) = 0 and Var(Z ) = 1. Then E(Y ) = µ and
Var(Y ) = σ 2 .
Y −µ
σ
∼ N(0, 1).
Example
If X is a normal random variable with parameter µ = 3 and
σ 2 = 9, find (a) P(2 < X < 5); (b) P(X > 0); (c) P(|X − 3| > 6).
Normal approximation to Binomial
Suppose X has a Binomial distribution with parameters n and
p. Then X can be written as X = I1 + · · · + In where Ii are
independent Bernoulli(p). Then X is a sum of independent and
identically distributed random variables. The distribution of X
can be approximated by normal distribution if n is large and p is
not extremely close to 0 and 1. Namely, for a < b
)
(
X − np
≤ b ≈ Φ(b) − Φ(a).
P a≤ p
np(1 − p)
Example: Normal approximation to Binomial
If n = 25 and p = 0.6, then
X − 15
13 − 15 .
13 − 15
P(X ≤ 13) = P( √
≤ √
) = P(Z ≤ √
) = 0.206,
6
6
6
where Z ∼ N(0, 1).
Gamma distribution
I
The Gamma distributed random variable T with
parameters α and 1 has the density function
f (t) =
1 α−1 −t
t
e
Γ(α)
0 < t < ∞.
which is an extension of Gamma function.
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The full Gamma distribution family is a scale
transformation of T . Define X = βT . Then
f (x; α, β) =
1
x α−1 e−x/β
Γ(α)β α
0 < x < ∞,
where α is a shape parameter and β is a scale parameter.
We will denote it as Gamma(α, β).
1.5
Gamma density plots
0.0
0.5
Density
1.0
alpha=1,beta=1
alpha=3,beta=1
alpha=2,beta=3
alpha=1,beta=3
0
2
4
x
6
8
Mean and variance
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If X ∼ Gamma(α, β), then
E(X ) = αβ and Var(X ) = αβ 2 .
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Exponential distribution: Suppose X ∼ Gamma(α, β). If
α = 1, then X has an exponential distribution with
parameter β. It has pdf
fX (x) =
1
exp(−x/β).
β
Memoryless Property
I
The probability that an instrument survives for at least s + t
hours given that it has survived t hours, is the same as the
intial probability that it survives for at least s hours. In
another words, if the instrument is alive at time t, the
distribution of the remaining lifetime is the same as the
orginal lifetime distribution.
P(X > s + t|X > t) =
exp(−(s + t)/β)
P(X > s + t)
=
P(X > t)
exp(−t/β)
= exp(−(s + t)/β) = P(X > s).
Hazard rate function
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Let X be a positive random variable represeting the lifetime
of some item.
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Hazard rate function λ(t) = f (t)/F̄ (t) represents the
conditional probability that an item will fail for an additional
time dt, given that it survivied until time t. i.e.,
P(X ∈ (t, t + dt), X > t)
P(X > t)
f (t)dt
≈
F̄ (t)
P(X ∈ (t, t + dt)|X > t) =
where F̄ (t) = 1 − F (t).
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If X follows an exponential(λ) distribution, then the hazard
rate function is λ(t) =
1
λ
exp(−t/λ)
exp(−t/λ)
= λ1 .