# Summary of Distributions

```Type
Description
Finite
Random
Variable
Finite discrete-it can
take on only finitely
many possible
values (ex: X =
0,1,2, or 3). In this
case you can list all
possible values
Probability Distribution
Func.
Probability mass function
(p.m.f.), fX (x) where
fX (x) = P(X = x)
= height of p.m.f.
 0  f X ( x)  1
 sum of p.m.f. values = 1
Cumulative Distribution
Func.
Cumulative density
function (c.d.f.),
Probability mass function
(p.m.f.), fX (x) where
fX (x) = P(X = x)
In this case, a histogram
associated with P( X  x)
Cumulative density
function (c.d.f.),
Probability Density
function (p.d.f.), The
value of the p.d.f., fX (x)
Cumulative density
function (c.d.f.),
FX  x   P( X  x)
 non-decreasing
 max value of 1
 step function
Expected Value, &micro;
Variance, 𝜎 2
Std Dev = 𝜎
E( X ) 
𝑉(𝑋) =

∑(𝑥 − 𝜇)2 𝑓𝑋 (𝑥)
x  P( X  x)
all x

 x  f X ( x)
all x
Binomia
l
Random
Variable
Continu
ous
Random
Variable
Binomial Setting:
1.You have n
repeated trials of an
experiment.
2. On a single trial,
there are two
possible outcomes,
success or failure.
3.The probability of
success, p , is the
same from trial to
trial.
4.The outcome of
each trial is
independent.
Continuous-if the
possible values form
an entire interval of
numbers (ex: any
positive number)
Expone
ntial
Random
Variable
Exponential random
variables are
continuous random
variables and
usually describe the
waiting time
between consecutive
events.
Uniform
Random
Variable
If X is uniform on
the interval [a,b]
then we have a
continuous uniform
random variable
 P( X  x) .
 The value of fX (x) is
simply the height of the
density curve at the value of
x.
 fX (x) &gt; 0 
FX  x   P( X  x)
E( X )  n  p
𝑉(𝑋) =
𝑛𝑝(1 − 𝑝)
 non-decreasing
 max value of 1
 step function
FX ( x)  P( X  x)
P(a  X  b)  FX (b)  FX (a)
𝐸(𝑋) =
𝑉(𝑋) =
∫(𝑥 − 𝜇)2 𝑓𝑋 (𝑥)𝑑𝑥
∞
∫ 𝑥 ∙ 𝑓𝑋 (𝑥)𝑑𝑥
−∞
P( X  x)  1  FX ( x)
P( X  x)  0 for all x
 non-decreasing
 max value of 1
 continuous function,
usually
Probability Density
function (p.d.f.),
Cumulative density
function (c.d.f.),
E X  
𝑉(𝑋) = 𝛼 2
0 if x  0
0 if x  0
FX ( x)  
f X ( x)   1  x / 
if 0  x
1  e  x /  if 0  x
 e


Probability Density
function (p.d.f.),
0
 1

f X ( x)  
b  a
0
if x  a
Cumulative density
function (c.d.f.),
if x  a
0
xa
if a  x  b F X ( x)  
if a  x  b
b  a
if x  b
if x  b
1
E( X ) 
ba
2
𝑉(𝑥)
(𝑏 − 𝑎)2
=
12
```