WP WP-34S, Owner'S Manual

Page: 15 / 103

WP 34S Owner‘s Manual
Edition 2.2 Page 15 of 103
STATISTICAL DISTRIBUTIONS, PROBABILITIES ETC.
You will find a lot of statistics in your WP 34S, going far beyond the Gaussian distri-
bution. Many preprogrammed functions are implemented here for the first time in an
RPN calculator
– we packed all in what we always had missed. All of these functions
have a few features in common:

Discrete statistical distributions (e.g. Poisson, Binomial) are confined to integers.
Whenever we sum up a probability mass function ( pmf
4
) to get a cumulated
distribution function ( cdf ) we start at . Thus,
 
m P n p m F
m
n
 


0
) ( ) ( .

Whenever we integrate a function, we start at the left end of the integration inter-
val. Thus, integrating a continuous probability density function ( pdf ) to get a
cdf typically works as
   
x P d f x F
x
 

 
 
) ( .

Typically,
F
starts with a very shallow slope, becomes steeper then, and runs out
with a decreasing slope while slowly approaching 100%. Obviously you get the
most precise results on the left side of the
cdf using P . On its right side, howev-
er, the ―error probability‖
Q = 1
– P
is more precise: since P comes very close
to 100% there, you may see 1.0000 displayed while e.g. P = 0.99996 in reality.

On your WP 34S, with an arbitrary cdf named XYZ you find the name XYZ
-1
for its
inverse and XYZ
P
for the pdf or pmf , unless stated otherwise explicitly.

For calculating confidence limits for the ―true value‖ based on a sample evalua-
tion, employing a particular confidence level (e.g. 95%), you must know your ob-
jective:
o
Do you want to know the upper limit, under which the
―true value― will lie with a
probability of 95%? Then take 0.95 as the argument of the inverse cdf to get
said limit, and remember there is an inevitable chance of 100%
– 95% = 5%
for the ―true value‖ being greater than it.
o
Do you want an upper and
a lower limit confining the ―true value‖? Then there
is an inevitable chance of 5% / 2 = 2.5% for said value being less than the
4
In a nutshell, discrete
statistical distributions deal with ―events‖ governed by a known mathematical
model. The pmf then tells the probability to observe a certain number of such events, e.g. 7. And the
cdf tells the probability to observe up to 7 such events, but not more.
For doing statistics with continuous statistical variables
– e.g. the heights of three-year-old toddlers –
similar rules apply: Assume we know the applicable mathematical model. Then the respective cdf
tells the probability for their heights being less than an arbitrary limit value, for example less than 1m.
And the corresponding
pdf
tells how these heights are distributed in a sample of let‘s say 1000 chil-
dren of this age.
WARNING: This is a very coarse sketch of this topic only
– please turn to textbooks about statis-
tics to learn dealing with it properly.
The terms pmf and pdf
translate to German „Dichtefunktion― or „Wahrscheinlichkeitsdichte―,
cdf to
„Verteilungsfunktion― or „Wahrscheinlichkeitsverteilung―.