TableofAnalogies
How do we graph the distribution? Or what does the graph look like? What do we call the distribution function? f X ( x) Discrete Random Variable A discontinuous function that has “points of mass” I like to do needle plots (lollipop plots) to make those points of mass look like they have mass, but technically, the needles (the lollipop stick) aren’t part of the plot. If you connect the dots, you are doing it to help your brain—to make a pattern easier to see—but it isn’t mathematically proper to do so. probability mass function (PMF) probability distribution function P ( X  x) Continuous Random Variable A continuous function A continuous curve Dataset A histogram or a boxplot or stem‐and‐leaf plot or some other variant of these plots The y‐axis could be frequency or relative frequency Another word for relative frequency is proportion probability density function (PDF) probability distribution function lim P( x  h  X  x  h) =
empirical distribution The relative frequency of x . The proportion of observations in the dataset that equal x . Number of times x occurs in dataset/Number of observations in dataset n/a h 0
lim P( x  h  X  x  h) h 0
What makes f X ( x) valid? 1.
2.
0  f X ( x)  1 , and f
X
( x)  1 x
1.
f X ( x)  0 , and 
2.

f X ( x)dx  1 
P ( X  x) Some number between 0 and 1, depending on what x is. 0 always Mean Symbols commonly used are EX , E ( X ) , and  , and the definition of the quantity is Two symbols commonly used are x and Symbols commonly used are EX , E ( X ) , and  , and the definition of the xn , and the definition of the quantity is quantity is 1 n

x
xi n i 1
EX  xf X ( x)dx EX   xf X ( x)
{ x}


Some number between 0 and 1, depending on what x is. 
TableofAnalogies
Variance Discrete Random Variable Symbols commonly used are Var ( X ) , Continuous Random Variable Symbols commonly used are Var ( X ) , V ( X ) , and  2 , and the definition of the V ( X ) , and  2 , and the definition of quantity is V ( X )   ( x  EX ) 2 f X ( x) { x}
the quantity is  ( x  EX )
Symbol used is s 2 , and the definition of the quantity is s2 

V (X ) 
Dataset 2
f X ( x)dx 1 n
 ( xi  x )2 n  1 i 1

Cumulative distribution function (CDF) FX (a)  P( X  a) P ( a  X  b) 
{ x  a}
f X ( x) X
( x)dx The proportion of observations in the dataset that are less than or equal to x . Percentiles. The proportion of observations in the dataset that fall in the interval [ a, b] X
( x)dx The proportion of observations in the dataset that fall in the interval (a, b) a

f X ( x)dx 
x b
f
xa
P ( a  X  b) 
b
X
{ a  x  b}
( x) f
a
f X ( x) b
f
a
 vs.  Important distinction No difference Important distinction