Continuous Random Variables
General:
FX  x   Pr  X  x  
f x  
x
 f t dt

dFX  x 
dx
E X    xf  x dx  
Var  X     x   2 f  x dx   2
 
Var  X   E X 2   2
StdDev X   Var  X   
Uniform on the interval (a, b).
1
a xb
b  a 
0
elsewhere
f x  
FX  x   0
FX  x  
x  a 
b  a 
FX  x   1
E X  
xa
a xb
xb
ab
2
Var  X  
b  a 2
12
1
Beta with parameters α and β: This random variable is often used to model proportions
or percentages e.g. the proportion of impurities in a batch of chemical.
     1
1  x  1 0  x  1
x
  
elsewhere
0
f x  
FX  x  does not have a nice closed form in general.
E X  

 
Var  X  

    1   2
Exponential with parameter β: This random variable is often used to model waiting
times or times between occurrences of events generated by a Poisson process.
f  x    e  x
0
FX  x   0
0 x
elsewhere
x0
FX  x   1  e   x
E X  
0x
1

1
Var  X    
 
2
Gamma with parameters α and β:
f x  
   1  x
x
e
0 x
 
0
elsewhere
FX  x  does not have a nice closed form in general.
E X  


Var  X  

 2
2
Normal with parameters μ and σ:
x   
2
1
e
2 
f x  

2
2
 x
F X  x  does not have a nice closed form in general.
EX   
Var  X    2
Log Normal with parameters μ and σ: Y is a continuous random variable taking on
positive values. If log(Y) is a Normal random variable with parameters μ and σ, then Y is
a Log Normal random variable. Or if X is a Normal random variable with parameters μ
and σ, then Y=eX has a Log Normal distribution.
log( y )  
2
f  y 
1
e
2 y
0

2
2
0 y
elsewhere
FY  y  does not have a nice closed form in general.
2
EY   e   2
2
 1
Var Y   e 2  2 [e ]
2
3