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 In other words, a random variable is like a function that takes in the
real world and spits out a number in its range.
 Ex. X = the # on a dice roll (Range: 16)
 W = X^{2} (1,4,9,16,25,36)
 Y = the # of ppl who are awake (0~70)
 Z = the # of ppl born today (nonneg. Int)
 V = the kg. of grain harvested this year in Canada (Real no.)
 There is some probability that the random variable will equal every
value in its range.

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 Note that for every random variable X,
 å_{x} P(X=x) =1,
 where x ranges over all possible values that X can take.
 All the possible values that X can take and the probability that X is
any one of those values is called the distribution of X.

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 Let X be the sum of two dice and Y be the minimum.

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 The distribution of (X,Y) is called
 the joint distribution of X and Y:
 P(x,y) = P(X=x,Y=y),
 satisfying
 P(x,y)¸ 0 and å_{all (x,y)}
P(x,y) = 1.

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 We can build a probability distribution
 table for the X and Y in the previous example:

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 Question:
 Suppose you know the distribution of X and the distribution of Y
separately, does this determine their joint distribution?

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 Question:
 Suppose you know the distribution of X and the distribution of Y
separately, does this determine their joint distribution?
 Answer:
 It does not …

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 Random Variables X and Y have the same or identical distribution if they
have the same range and for every value x in their range
 P(X=x) = P (Y=x).

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 Having the same distribution is not enough. If we consider the function
analogy, two random variables are equal if they spit out the same number
for every status of the real world.
 Of course if they are equal, they will have the same distribution.

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 If X and Y have the same distribution then so do g(X) and g(Y), for any function
g. For example:
 X^{2} has the same
distribution as Y^{2}.

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 Random variables X and Y defined over the same outcome space are equal,
written X=Y, iff
 X= Y for every outcome in W.
 In particular, P(X=Y) = 1.

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 Example:
 Suppose we have a box with tiles each having the same area A.

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 The probability that X and Y satisfy some condition is the sum of P(x,y)
over all pairs (x,y) which satisfy that condition.

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 We can add two random variables:
 If X is the no. on a 6 sided die
 Y is {3 if Professor Mossel is wearing a blue shirt today, 0 if not}
 Then X+Y is then sum of the two individual random variable values.
 How many ways can X+Y = 0? 1? 4? 8?

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 Distribution of any function of X and Y can be determined from the joint
distribution:

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 If the marginal distribution of X and the conditional distribution of Y
given X=x are known, then we can find the joint distribution of X and Y:
 P(X=x, Y=y) = P(X=x) P (Y=yX=x).

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 This property does not always hold!
 P(X=x, Y=y) ¹ P(X=x) P(Y=y)
 for general X and Y.

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 The distribution of (X_{1},X_{2}, …, X_{n}) is
called the joint distribution of X_{1},X_{2}, …, X_{n}
:
 P(x_{1},x_{2},…,x_{n}) =P(X_{1}=x_{1},X_{2}=x_{2},
…, X_{n}=x_{n} ),
 satisfying
 P(x_{1},x_{2},…,x_{n})¸ 0 and
 å_{all (x}_{1}_{,x}_{2}_{,…,x}_{n}_{)}
P(x_{1},x_{2},…,x_{n}) = 1.

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 If X_{1},X_{2},
…, X_{n }are independent then
so are the random variables
 Y_{1} = f_{1}(X_{1}) , … ,Y_{n} = f_{n}(X_{n}).
 For any functions f_{1}, …, f_{n} defined on the range
of X_{1}, …, X_{n}.

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 Example:
 X_{1}, X_{2}, X_{3},X_{4},X_{5}
are the numbers on 5 dice, and f(x) = 0 if x is even and f(x) = 1 if x
is odd. If we let Y_{i} = f(X_{i}).,Then Y_{1},Y_{2},Y_{3},Y_{4},Y_{5}
are independent.

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 Let’s consider the 5 dice again.

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 If X_{1},…,X_{n} are independent.
 And B_{1},…,B_{k} are disjoint subsets of positive
integers such that
 B_{1} [ B_{2} … [ B_{k} = {1,2,…,n}.
 Then the random vectors
 (X_{i} : i 2 B_{1}), (X_{i} : i 2 B_{2})
,…, (X_{i} : i 2 B_{k})
 are independent.

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 If X_{1},X_{2},
…, X_{5 }are independent then
so are the random variables
 Y_{1}=X_{1} +X_{3}^{2}; Y_{2} =X_{2} X_{4
}; Y_{3} =X_{5}

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