Random V ector

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# Random V ector - PowerPoint PPT Presentation

Random V ector. Tutorial 6, STAT1301 Fall 2010, 02NOV2010, MB103@HKU By Joseph Dong. Recall: Cartesian Product of Sets. Two discrete sets. Two Continuous sets. Recall: sample Space of A Random Variable.

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### Random Vector

Tutorial 6, STAT1301 Fall 2010, 02NOV2010, MB103@HKUBy Joseph Dong

Recall: Cartesian Product of Sets

Two discrete sets

Two Continuous sets

The Making of a Random Vector as Joint Random Variables: A Crash course of Latin Number Prefixes
• Uni-variate : 1 random variable
• Bi-variate : 2 random variables bind together to become a 2-tuple random vector like
• Tri-variate : 3 random variables bind together to become a 3-tuple random vector like
• ……
• n-variate : n random variables bind together to become a 3-tuple random vector like
• You can even have infinite-dimensional random vectors! Unimaginable!
Random Vector as a function itself:
• How to distribute total probability mass 1 on the sample space of the random vector?
• Is this process completely fixed?
• If not fixed, is this process completely arbitrary?
• If neither arbitrary, what are the rules for distributing total probability mass 1 onto this state space?
• “Marginal PDF/PMF” imposes an additive restriction.
• There is a lot to discover here…
Independence among random Variables
• Recall: What are independence among events?
• Q: What does a random variable do to its state space?
• It partitions the state space by the atoms in the sample space!
• is an atom in the sample space and is a block in the state space.
• is a union of atoms in the sample space and is a union of blocks in the state space.
• We can talk about whether and are independent
• because they mean two events: and
• We can talk about whether and are independent
• because they mean two events: and
• Goal: Generalizethis connection to the most extent: Establish the meaning of independence between whole random variables and .
Two random Variables are independent if…
• Each event in the state space of is independent from each event in the state space of .
• Further, this is true if each atom in the state space of is independent from each atom in the state space of .
• How many terms are there if you expand ?
• One more equivalent condition:
Independence of Continuous Random Variables
• Previous picture deals with the discrete random variables case.
• Two continuous random variables and are independent if
• or/and
• or/and
Determine independence solely from the Joint distribution
• If you are only given the form of or how do you know that and are independent?
• Check if or can be factorized into a product of two functions, one is solely a function of , the other solely a function of .
• , are independent
• Clearly vice versa
• Pf.
Expectation vector
• Define the expecation of a random vector as
• It’s still the (multi-dimensional) coordinate of the center of mass of the joint sample space (Cartesian product of each individual sample spaces).
• E.g. The center of mass of a massed region in a plane.
• E.g. The center of mass of a massed chunk in a 3D space.
• For the expectation of a scalar-valued function of random vector can be computed using Lotus as:
• Expectation of independent product: If and are independent, then
• Pf.
• MGF of independent sum: If and are independent, then
• Pf.
A short Summary for Independent Random Variables
• First of all, the bedrock (joint sample space) must be a rectangular region.
• Refer to the problem on Slide 9 of Tutorial 2.
• Then you must be careful to equip each point in that region with a probability mass (for discrete case) or a probability density (for continuous case).
• The rules are
• Total probability mass is 1
• The probability mass/density distributed on each column must sum/integrate to the that column’s marginal probability mass/density.
• The probability mass/density distributed on each row must sum/integrate to the that row’s marginal probability mass/density.
• Your goal is to make either of the following true at every point in the joint space
• Intuition: there cannot be cave-like vertical openings of the density surface over the joint sample space.
• Rigorous definition:
• There exists density function everywhere on the joint sample space.
Joint CDF
• Check more properties of joint CDF and the relationship between joint CDF and joint PMF/PDF in the review part of handout.