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Uniform Random Variables are used where each outcome of an event is equally probable. For
p
U
(
u
)
=
P
(
U
≤
u
)
{\displaystyle p_{U}(u)=P(U\leq u)}
U
{\displaystyle U\,}
[
a
,
b
]
{\displaystyle [a,b]\,}
p
U
(
u
)
{\displaystyle p_{U}(u)\,}
(
a
≤
u
≤
b
)
{\displaystyle (a\leq u\leq b)}
(
a
≤
u
≤
b
)
{\displaystyle (a\leq u\leq b)}
p
U
(
u
)
=
1
b
−
a
{\displaystyle p_{U}(u)={\frac {1}{b-a}}}
∫
−
∞
∞
f
X
(
x
)
δ
x
{\displaystyle \int _{-\infty }^{\infty }f_{X}(x)\delta x}
=
1
{\displaystyle =1\,}
f
X
(
x
)
{\displaystyle f_{X}(x)\,}
=
{
1
b
−
a
a
<
x
<
b
0
else
{\displaystyle ={\begin{cases}{\frac {1}{b-a}}&a<x<b\\0&{\mbox{else}}\\\end{cases}}}
F
X
(
x
)
{\displaystyle F_{X}(x)\,}
=
{
0
x
<
a
x
−
a
b
−
a
a
≤
x
<
b
1
x
≥
b
{\displaystyle ={\begin{cases}0&x<a\\{\frac {x-a}{b-a}}&a\leq x<b\\1&x\geq b\\\end{cases}}}
h
X
(
x
)
{\displaystyle h_{X}(x)\,}
=
?
?
?
{\displaystyle =???\,}
M
X
(
x
)
{\displaystyle M_{X}(x)\,}
=
e
x
b
−
e
x
a
x
(
b
−
a
)
{\displaystyle ={\frac {e^{xb}-e^{xa}}{x\left(b-a\right)}}}
![{\displaystyle [a,b]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23cb97ebba2cd3175f9a77446963c1849fc353ee)
