VARIABLES
Measurements RANDOM and observations are called
random variables. Each results from an
outcome of an experiment, so:
Denition: Random variable X is real
function on sample space S.
Start with discrete r.v. RX is nite or
countable.
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For any real x RX , the set {X = x}
is an event and so has a probability.
Denition: The probability mass function of X is the function
f (x) = P (X = x)
Properties.
f (x) 0
xRX
f (x) = 1
P (X A ) =
xA f (x)
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Example: Benfords law
First observed by Frank Benford working at GE Research
Laboratories in 1920s. In many real-world datasets the
rst digit of a number follows a distribution:
First Digit
Probability
First Digit
Probability
1
0.301
6
0.067
2
0.176
7
0.058
3
0.125
8
0.051
4
0.097
9
0.046
5
0.079
Consider a market capital of a randomly picked publicly
traded company. Let X = rst digit of market capital
amount.
According to Benfords law P (X = 1) = 0.301.
References:
Ted Hill, The rst-digit phenomenon, American Scientist, 86(1996), pp.358-363.
Fraud detection: http://www.nigrini.com/
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Origin of Benfords Law
Leading digit X of positive numbers is the rst non-zero
number in its decimal representation. So the leading
digits of .0023, 2.23, and .234 are all 2.
Benford wasnt the rst to observe this phenomenon.
Astronomer Simon Newcomb, looking at tables of logarithm, concluded that leading distribution of real data
was not typically uniformly distributed. In 1881 he gave
an argument to show that the mantissas of the logarithms of the numbers should be uniform
P (X = x) = f (x) = log(x + 1) log(x) for x = 1, 2, .., 9
x
f ( x)
1
.301
2
.176
3
.125
4
.097
5
.079
6
.067
7
.058
8
.051
4
9
.046
Example 2.3-1, p 107. Fair coin tossed
three times. X = # heads.
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P (X = x ) =
x
x
0
1
2
3
13
2
f (x)
1/8
3/8
3/8
1/8
sum = 1
5
Example 2.3-2, p 108. Toss fair coin repeatedly until rst head. X = # tosses.
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11
1x
P (X x) = =
=
...
22
22
2
x 1 times
Verify
1x
x=1 2
= 1 using sum of ge-
ometric series.
Denition: The (cumulative) distribution function of X is the function
F (x) P (X x) =
f (t)
tx
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Use either f (x) or F (x) to nd probabilities. For Example 2.3-2, nd P (X =
3, 4, 5).
P (X = 3, 4, 5) =
=
f (x)
x=3,4,5
13
2
14
15
7
+
+
=
2
2
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OR rst nd
1t
1x
F (x) =
=1
2
t=1 2
x
and recognize that
P (a X b) = F (b) F (a 1)
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Then
P (X = 3, 4, 5) = P (X 5) P (X 2)
15
12
=1
(1
)
2
2
1
7
1
=
=
4 32
32
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Expectations
Expectation (or expected value or mean
value or mean) is the statistical equivalent of an average. Multiple each possible value of a r.v. by its probability
mass function.
E [X ] =
xf (x)
x
More generally if u is a real function
then u(X ) is also a random variable and
its expectation is
E [u(X )] =
u(x)f (x)
x
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E [X ] is often denoted by X or simply if no ambiguity. Expectation has
many properties. It is the centre of
mass of the probability mass function.
Also, if X1, X2, . . . , XN are observations
made of a measurement X with mean
then the long run average value is
approximately E [X ] for large N , that is
X1 + X2 + . . . + XN
N
This is called the Law of Large Numbers or more colloquially, the law of averages.
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Find the expected value of Lotto. Earlier
x
0
1
2
3
4
5
6
f (x)
0.436
0.413
0.132
0.0177
0.000969
0.0000184
0.0000000715
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E [X ] =
xf (x) = 0.734
x=1
So on average you lose about 27 cents for each
dollar wagered.
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Variance and Standard Deviation
Denition: Variance of X is given by
VarX 2 = E [(X )2] =
(x)2f (x)
x
Denition: Standard deviation of X
is , the positive square root of the
variance.
Equivalent computational formula
2 = E [X 2] 2
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We will see shortly that is a measure
of the variability of a random variable
and plays for X a similar role that s
plays for a data set.
Exercise 2.3-5. To be done in class.
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