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Random Variable
Tutorial 3
STAT1301 Fall 2010
05OCT2010, MB103@HKU
By Joseph Dong
• A random variable is a function
𝑋: Ω ∋ 𝜔 ↦ 𝑋 𝜔 ∈ 𝑇
• 𝑇 = 𝑋 Ω is called the sample space. It is usually a
numbers set, i.e., a subset of ℝ or ℂ.
• A random variable is deterministic. The randomness does
not reside in the random variable 𝑋. The randomness
resides in the state space Ω, and is carried by 𝑋 over to
the sample space.
The Definition
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What’s the random experiment?
What are the possible outcomes of the random experiment?
What is the state space?
What are the possible values of money you can win?
What is the sample space?
What is the probability measure ℙ on the state space?
What is the probability measure ℙ𝑋 on the sample space?
What is the distribution function 𝐹𝑋 ⋅ on the sample space?
Handout Problem 1
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• R.E. = Tossing two dice simultaneously to observe 2
digits.
• The possible physical outcomes are (1,1), (1,2), ……,
(6,6). There are 36 of them.
• The State Space is the set of all physical outcomes
(states): {(1,1), …, (6,6)}.
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• The possible values of money I can win are 9, -10, and 0.
• The Sample Space is the set of all possible numerical
outcomes I can win: {9, -10, 0}
5
• The (discrete) Probability measure ℙ on the
state space Ω is a function that evaluates
1
every singleton subset of Ω to and that
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evaluates every composite subset according
to (i) the countable additivity axiom in
Kolmogorov’s definition of Probability; and
(ii) how it evaluates on the singleton subsets.
• For example, ℙ will evaluate the composite
1
1
subset {(1,1), (1,2)} to + because
36
36
• (i) the singleton sets {(1,1)} and {(1,2)} are
disjoint and therefore ℙ 1,1 ∪ 1,2 =
ℙ 1,1 + ℙ 1,2 ; and
• (ii) ℙ 1,1
= ℙ 1,2
=
1
.
36
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• The (discrete) Probability measure ℙ𝑋 on
the Sample Space 𝑋 Ω is a function
• that evaluates the singleton subsets of 𝑋 Ω ,
6 6 24
{9}, {-10}, {0} to , , respectively, and
36 36 36
• that evaluates every composite subset
according to (i) the countable additivity
axiom in Kolmogorov’s definition of
Probability; and (ii) how it evaluates on the
singleton subsets.
• For example, ℙ𝑋 will evaluate the composite
6
24
subset {9, 0} to + because (i) the
36
36
singleton sets {9} and {0} are disjoint and
therefore ℙ𝑋 9,0 = ℙ𝑋 {9} + ℙ𝑋 0 ; and (ii)
6
24
ℙ𝑋 9 = and ℙ𝑋 0 = .
36
36
• Q: How is ℙ𝑿 linked to ℙ by 𝑿?
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Choice of R.E. is flexible
• R.E. = Tossing two dice simultaneously to observe the
sum of the 2 digits produced.
OR
• R.E. = Tossing two dice simultaneously to observe the
value of money you can win based on the sum of the 2
digits produced.
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• Conclusion: Since the 3 spaces above have consistent
probability measure, any one can be used as our state space or
sample space, depending on your choice.
• They are just different representations.
• The consistency across the spaces are guaranteed by the defining
nature of the random variable between them.
• Make sure you use the right probability measure for the
sample/state space you work on.
• The choice of a good sample space is an art. A good choice of
sample space—and accordingly its probability measure—can
greatly simplify the solution process.
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• Because of the “deterministic” random variable 𝑋 and the
“random” state space Ω, the sample space 𝑇 = 𝑋 Ω as the
combination of the two is also “random”.
• Therefore we usually have each of Ω and 𝑋 Ω endowed with
a probability measure, for the depiction of their randomness.
• As usual, we use ℙ to denote the probability measure on the
state space Ω.
• We use a subscripted ℙ𝑋 to symbolize the probability measure
of the sample space 𝑋 Ω .
• ℙ and ℙ𝑋 must be consistent because 𝑋 is deterministic.
A Short Wrap-up
Ω & ℙ vs 𝑋 Ω & ℙ𝑋
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Going Visual
Random
Variable/Function
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Sample Space Illustrated
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• Hint: If you understand our discussion of Problem 1, you
should immediately know an example for Problem 6.
• Also try to find a different kind of example.
Handout Problem 6
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• Cumulative
Distribution
Function
• Distribution Function 𝑭𝑿 𝒙
• Probability
Density
Function
• Density Function 𝒇𝑿 𝒙
• Probability
Mass
Function
• ℙ𝑋 𝑋 ≤ 𝑥
• ℙ𝑋 𝑋 = 𝑥
• Probability Function
•
Distribution
Nomenclature
ℙ𝑋 𝑥≤𝑋<𝑥+𝑑𝑥
lim
𝑑𝑥
𝑑𝑥→0+
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• The term distribution function is used in the
mathematical literature for never-decreasing functions of
𝑥 which tend to 0 as 𝑥 → −∞, and to 1 as 𝑥 → ∞.
Statisticians currently prefer the term cumulative
distribution function, but the adjective “cumulative” is
redundant.
• A density function is a non-negative function 𝑓 𝑥 whose
integral, extended over the entire 𝑥-axis, is unity.
• The integral from −∞ to 𝑥 of any density function is a
distribution function.
—William Feller: An Introduction to Probability Theory and Its
Applications (1950) Volume I, page 179: “Note on Terminology.”
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• Handout Problem 3
• Hint: 𝑎 + 𝑏
𝑛
=?
• Handout Problem 4
• Hint: Just routine calculation.
Handout Problem 3 and 4
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• We only define distribution function from ℝ to [0,1].
Therefore, strictly speaking, only real-valued random
variables can have a distribution function defined for its
sample space.
• A distribution function 𝐹𝑋 ⋅ is just an alternative way
besides the probability measure ℙ𝑋 to depict randomness.
• Relationship: 𝑭𝑿 ⋅ ≔ ℙ𝑿 (𝑿 ≤⋅)
• Still with Problem 1, what’s the distribution function
𝐹𝑋 ⋅ on the sample space {-10, 0, 9}?
Distribution Function
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• Draw a graph for each question to show the random
variable, the state space, the sample space, and the
probability mass function on the sample space.
Handout Problem 2
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• This time the state space is the interval 0,1 on the real
line.
• Try to use use the state space and the sample space as the
two ordinates of a Cartesian plane, and draw the graph of
the random variable on that coordinated plane.
Handout Problem 5
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• Hint: Try to
understand
“Expectation is the
coordinate of the
center of mass”
Handout Problem 7
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