Download UNIFORM The uniform distribution is useful in representing random

SOME SIMPLE PROBABILITY DENSITY FUNCTIONS (PDF)
f ( x) =
UNIFORM
1
β −α
α ≤x≤β
The uniform distribution is useful in representing random variables which have
known upper and lower bounds and which have equal likelihood of occurring
anywhere between these bounds.
That is, if you know nothing else about the relative likelihood of a random
variable, aside from its upper and lower bounds, then the uniform distribution
is appropriate -- it makes no assumptions regarding preferential likelihood of
the random variable since all possible values are equilikely.
f ( x)
E[ X ] =
α +β
2
0.25
( β − a)2
Var[ X ] =
12
0
1
2
3
4
5
6
7
8
9
x
EXPONENTIAL
E[T ] =
⎧λ e − λt , for t ≥ 0
fT (t ) = ⎨
⎩ 0, otherwise
1
λ
Var[T ] =
1
λ2
Suppose that the time-to-failure, T, of a clay barrier has the probability density
function;
⎧λ e − λt , for t ≥ 0
fT (t ) = ⎨
⎩ 0, otherwise
0.015
0.020
where λ = 0.02/year, then what is the probability that T exceeds 100 years?
∞
0.010
∫
100
fT (t ) dt =
∫ λe
100
0.005
= e −100 λ = e −100(0.02) = e −2
= 0.1353
P[ T > 100 ]
0
fT (t)
P[ T > 100 ] =
∞
0
50
100
t (years)
150
200
− λt
dt
NORMAL
The best known Probability Density Function is the
Normal or Gaussian distribution.
Let X be a normally distributed random variable
with mean and standard deviation given by µ X and σ X .
In this case the PDF is given by:
2
⎧
⎛
⎞
1
⎪ 1 x − µ X ⎫⎪
exp ⎨− ⎜
f X ( x) =
⎟ ⎬
2
σ
σ X 2π
X
⎠ ⎭⎪
⎪⎩ ⎝
E[ X ] = µ X
Var[ X ] = σ X2
NORMAL DISTRIBUTION
µ X = 100 σ X = 50
f X ( x)
Mean,median and mode
The area under the
distribution is unity
inflection points
are at µ ± σ
x