Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
RANDOM VECTOR Tutorial 6, STAT1301 Fall 2010, 02NOV2010, MB103@HKU By Joseph Dong RECALL: CARTESIAN PRODUCT OF SETS Two discrete sets Two Continuous sets 2 RECALL: SAMPLE SPACE OF A RANDOM VARIABLE 3 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- Num. 1 2 3 4 5 Sexa- Septi6 7 Octo- Novem- Deca- 8 9 10 4 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… 5 INDEPENDENCE AMONG RANDOM VARIABLES • Recall: What are independence among events? • ℙ • ℙ ⋅ℙ Q: What does a random variable do to its state space? • It partitions the state space by the atoms in the sample space! • is a block in the state space. is an atom in the sample space and • , is a union of atoms in the sample space and in the state space. • We can talk about whether and We can talk about whether ∈ • because they mean two events: • is a union of blocks are independent and • because they mean two events: • , and ∈ are independent and Goal: Generalize this connection to the most extent: Establish the meaning of independence between whole random variables and . 6 TWO RANDOM VARIABLES ARE INDEPENDENT IF… • Each event in the state space of is independent from each event in the state space of . ℙ • ∈ , ∈ ∀ ⊂ ℙ ∈ ℙ ∈ Ω ,∀ ⊂ Ω , Further, this is true if each atom in the state space of is independent from each atom in the state space of . ℙ , ∀ ∈ ℙ ℙ Ω ,∀ ∈ Ω • How many terms are there if you expand ? • One more equivalent condition: ℙ , ∀ ∈ ℙ ℙ Ω ,∀ ∈ Ω , , 7 INDEPENDENCE OF CONTINUOUS RANDOM VARIABLES • Previous picture deals with the discrete random variables case. • Two continuous random variables ∈ , • ℙ • • , , , , ∈ ℙ ∈ and ℙ are independent if ∈ ,∀ ⊂ Ω ,∀ ⊂ , ∀ ∈ Ω ,∀ ∈ Ω , ∀ ∈ Ω ,∀ ∈ Ω Ω or/and or/and 8 DETERMINE INDEPENDENCE SOLELY FROM THE JOINT DISTRIBUTION • If you are only given the form of ℙ and are independent? , or , , how do you know that • Check if ℙ , or , can be factorized into a product of two , functions, one is solely a function of , the other solely a function of . • ℙ , or , , ⟹ , are independent • Clearly vice versa • Pf. 9 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. 10 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 • ℙ • , , ℙ , in the joint space ℙ , 11 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 • ℙ , ∈ ∬ , ∈ everywhere on the joint sample space. , , ∀ ∈ 12 JOINT CDF , • Check more properties of joint CDF and the relationship between joint CDF and joint PMF/PDF in the review part of handout. 13 EXERCISE TIME 14