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

RANDOM VECTOR

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RECALL: CARTESIAN PRODUCT OF SETS

Two discrete sets Two Continuous sets

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RECALL: SAMPLE SPACE OF A RANDOM VARIABLE

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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!

Prefix Uni- Bi- Tri- Quadri- Quinti- Sexa- Septi- Octo- Novem- Deca-

Num. 1 2 3 4 5 6 7 8 9 10

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• 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…

RANDOM VECTOR AS A FUNCTION ITSELF:

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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: Generalize this connection to the most extent: Establish the meaning of independence between whole random variables and .

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• 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:

TWO RANDOM VARIABLES ARE INDEPENDENT IF…

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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

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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.

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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.

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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

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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.

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

JOINT CDF

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EXERCISE TIME