Transcript Cumulative Distribution Functions

Cumulative Distribution Functions
• The cumulative distribution function of a random
variable X, written, F, indicates the probability
that is at and to the left of each point x.
F x  P X  x
• CDFs are defined for both discrete and continuous
random variables.
Properties of a CDF
1. lim F x   0
x  
2. lim F x   1
x 
3. If a  b, then F a   F b  for any real
numbersa and b.
CDFs for continuous RVs
• If X is a continuous RV with density f, the
cumulative distribution function of X is
F x   P X  x  
x
 f t dt

for    x  
Special Properties of a CDF for a
Continuous RV
Pa  x  b   F b   F a  for real constantsa  b.
dFx 
f x  
, where thederivativeexists.
dx