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Manipulating Probability Expressions

Simplify the following expressions or say if they cannot be simplified. State which equation you used. See list of equations at bottom of unit 2, lecture 2 for names.

$$ \sum_X P(x) $$

$$ \sum_X P(x) = 1 $$

due to law of total probability

$$ \sum_X P(x \,|\, y) $$

$$ \sum_X P(x \,|\, y) = 1 $$

due to law of total probability

$$ \sum_Y P(x \,|\, y) $$

cannot be simplified

$$ \sum_Y P(x, y) $$

$$ \sum_Y P(x,y) = P(x) $$

due to definition of marginal

$$ \sum_Y P(x\, |\, y) P(y) $$

$$ \sum_Y P(x\, |\, y) P(y) = P(x) $$

due to marginalization of the conditional

$$ \frac{P(x, y)}{P(x)} $$

$$ \frac{P(x, y)}{P(x)} = P(y \,|\, x) $$

due to definition of conditional