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What is the difference between a random variable and a probability distribution? Provide an example

A random variable is only 1 variable of a certain world,

A probability distribution is a table of outcomes and their probabilities of the given RVs.

RV: Weather ( sunny cloudy rainy)

PD using the above RV is the table of probability the world is in any of the states in weather (sunny, cloudy, rainy)


In probability theory, what is a possible world? Provide an example.

A possible world can be any tuple in a JPD. Using the same example as question 1, all the provided examples in the probability distribution would be considered possible worlds


Explain, both in words and using math, the difference between P(H | E) and P (H.E)

P (h|e) is the conditional probability of h given evidence e, while P (H E) is the joint probability of h and e

P(H|E) = P(H ^ E) / P(E)
P(H,E) = P(H ^ E)


Consider the above joint probability distribution over the three random variables X, Y, Z. compute the marginal probability distributions for P(X), P(Y) and P(Z).

P(X) = [1, 0], P(Y) = [0,1], P(Z) = [1, 0]
Process: Marginalize each variable independently (e.g. we sum all columns where X is true, then all col where X is false; so we get that X is 1 when it’s true, 0 otherwise).


Name an operation that could be performed on a factor f that could decrease f's dimensionality and explain how much smaller the new factor would be

Summing out a variable, the new factor would be half the size, because you are removing a variable from the distribution, which means you are decreasing it by a multiple of 2.
(size of the table is 2^k, where k is the number of the variable assuming each variable with a domain of 2.)

Assigning a variable is another option, similar explanation


Name an operation that could be performed on a factor f that could increase f's dimensionality and explain how much bigger the new factor could be.

Multiplying factors (doing a join on common variables), The increased size would depend on how many common and uncommon variables there between the two factors. Increases by a factor of 2 for every uncommon variable from one factor to the other.


7. (Uncertainty) [8 points] Consider a node N in a Bnet. N's domain has size d. N has k parents, each with also domain size d. What is the size of the CPT for P(N|parentsOfN)? Is it true that the sum of all the entries in this CPT is equal to d? why or why not?

The size is d^k * (d-1)??

No because it is possible that there are more or less entries.


8. Uncertainty [6 points] How many different Markov Chains on k possible states there are, in which the agent is equally likely to start in any state and then it is stuck in the same state forever? Show the tables P(S0) and P(St+1|St) for one of such Chains when k = 3.

k! Since the agent is equally likely to start in any state and the markov chain only moves forward, therefore there at k! different Markov Chains