# Probability and Statistics ```Probability and Statistics
Probability vs. Statistics
• In probability, we build up from the
mathematics of permutations and
combinations and set theory a mathematical
theory of how outcomes of an experiment will
be distributed
• In statistics we go in the opposite direction:
We start from actual data, and measure it to
determine what mathematical model it fits
• Statistical measures are estimates of
underlying random variables
Basic Statistics
• Let X be a finite sequence of numbers (data
values) x1, …, xn
• E.g. X = 1, 3, 2, 4, 1, 4, 1 (n=7)
• Order doesn’t matter but we need a way of
allowing duplicates (“multiset”)
• Some measures:
– Maximum: 4
– Minimum: 1
– Median (as many ≥ as ≤):
• 1, 1, 1, 2, 3, 4, 4 so median = 2
– Mode (maximum frequency): 1
Sample Mean
• Let X be a finite sequence of numbers x1, …, xn.
• The sample mean of X is what we usually call the
average:
n
1
 X   xi
n i1
• For example, X = 1, 3, 2, 4, 1, 4, 1 , then μX=16/7
• Note that the mean need not be one of the data
values.
• We might as well write this as E[X] following the
notation used for random variables
Sample Variance
• The sample variance of a sequence of
data points is the mean of the square of
the difference from the sample mean:
n
1
2
Var(X)   (xi   X )
n i1
Standard Deviation
• Standard deviation is the square root of
the variance:
n
1
2
   (xi   X )
n i1
2
X
• σ is a measure of spread in the same units
as the data
•
•
•
•
•
•
•
X = 1, 2, 3
μ=2
σ2 = (1/3) ∙ ((1-2)2+(2-2)2+(3-2)2) = 2/3
σ ≈ .82
Y = 1, 2, 3, 4, 5
μ=3
σ2 = (1/5) ∙ ((1-3)2+(2-3)2+(3-3)2+(4-3)2+(53)2)
= 10/5
σ ≈ 1.4
Small σ Indicates “Centeredness”
•
•
•
•
•
•
•
•
•
•
•
X = 1, 2, 3
σ ≈ .82
Z = 1, 2, 2, 2, 3
μ=2
σ2 = (1/5) ∙ ((1-2)2+3∙(2-2)2+(3-2)2) = 2/5
σ ≈ .63
W = 1, 1, 2, 3, 3
μ=2
σ2 = (1/5) ∙ (2∙(1-2)2+(2-2)2+2∙(3-2)2) = 4/5
σ ≈ .89
(“Bimodal”)
Covariance
• Sometimes two quantities tend to vary in
the same way, even though neither is
exactly a function of the other
• For example, height and weight of people
Weight
Height
Covariance for Random Variables
• Roll two dice. Let X = larger of the two
values, Y = sum of the two values
• Mean of X = (1/36) &times;
(1&times;1
[only possibility is (1,1)]
+3&times;2 [(1,2), (2,1), (2,2)]
+5&times;3 [(1,3), (2,3), (3,3), (3,2), (3,1)]
+7&times;4 + 9&times;5 + 11&times;6)
= 4.47
Mean of Y = 7
How do we say that X tends to be large when Y
is large and vice versa?
Joint Probability
•
•
•
•
•
f(x,y) = Pr(X=x and Y=y) is a probability
Sums to 1 over all possible x and y
Pr(X=1 and Y=12) = 0
Pr(X=5 and Y=9) = 2/36
Pr(X≤5 and Y≥8) = 4/36
[(4,4), (4,5), (5,4), (5,5)]
Covariance of Random
Variables
• Cov(X, Y) = E[ (X − μX) ∙ (Y −μY) ]
• INSIDE the brackets, each of X and Y is
compared to its own mean
• The OUTER expectation is with respect to
the joint probability that X=x AND Y=y
• Positive if X tends to be greater than its mean
when Y is greater than its mean
• Negative if X tends to be greater than its
mean when Y is less than its mean
• But what are the units?
Sample Covariance
• Suppose we just have the data x1, …, xN
and y1, …, yN and we want to know the
extent to which these two sets of values
covary (eg height and weight). The sample
covariance is
n
1
(xi   X )(yi  Y )

n i1
• An estimate of the covariance
A Better Measure: Correlation
• Correlation is Covariance scaled to [-1,1]
 XY
Cov(X,Y )
E[(X   X )(Y  Y )]


 X  Y
Var(X)Var(Y )
• This is a unitless number!
• If X and Y vary in the same direction then
correlation is close to +1
• If they vary inversely then correlation is close
to -1
• If neither depends on the mean of the other
than the correlation is close to 0
Positively Correlated Data
Correlation Examples