Dr. Wyels, Prob/Stats I, Fall ’00
PDFs and PMFs
What is the total area under the bars of a relative frequency histogram associated with n
observations of a random variable of the continuous type?
Idea: as the number of observations (data
points) increases, and the number of classes
increases, the histogram should look more
and more like the (theoretical) distribution
from which the observations were made.
Wanted: a function f(x) satisfying
a) f ( x)  0, x  S
This part is left blank on
b)  f ( x) dx  1
the students’
S
worksheets: they
b
c) P (a  X  b)   f ( x) dx generate the three
a
criteria by adapting
what they know about
pmfs.
Dr. Wyels, Prob/Stats I, Fall ’00
Discrete Distributions
Probability Mass Function
a) f ( x)  0, x  S
b)
 f ( x)  1
xS
c) P( X  ui )  f (ui )
pmf/ pdf
(S is the
space of
possible
outcomes.)
Continuous Distributions
Probability Distribution Function
a)
b)
c)

mean

2 
variance
2 
Find  p (the 100pth percentile) by
median,
quartiles,
percentiles
 p is a number for which
Dr. Wyels, Prob/Stats I, Fall ’00
Discrete Distributions
Probability Mass Function
a) f ( x)  0, x  S
 f ( x)  1
pmf/ pdf
Continuous Distributions
Probability Distribution Function
a) f ( x)  0, x  S

(S is the
space of
possible
outcomes.)
b)
   xf ( x)
mean
   xf ( x) dx
 2   ( x   ) 2 f ( x)   x 2 f ( x)   2
variance
b)
xS
c) P( X  ui )  f (ui )
b
c) P (a  X  b)   f ( x) dx
a
S
xS
xS
f ( x) dx  1
S
xS
Find  p (the 100pth percentile) by taking the (n+1)pth
“data point”. (Interpolate between data points as
necessary.)
 2   ( x   ) 2 f ( x) dx   x 2 f ( x) dx   2
S
median,
quartiles,
percentiles
S
 p is a number for which the area under the pdf to the
p
left of  p is p. I.e. p  

f ( x) dx .