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Random V ector

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 V ector

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  1. Random Vector Tutorial 6, STAT1301 Fall 2010, 02NOV2010, MB103@HKUBy Joseph Dong

  2. Recall: Cartesian Product of Sets Two discrete sets Two Continuous sets

  3. Recall: sample Space of A Random Variable

  4. 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!

  5. 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…

  6. 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 .

  7. 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:

  8. 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

  9. 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.

  10. 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.

  11. 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

  12. Continuous Random Vector (or Jointly continuous Random Variables) • 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.

  13. Joint CDF • Check more properties of joint CDF and the relationship between joint CDF and joint PMF/PDF in the review part of handout.

  14. Exercise Time

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