Prove the following relationships algebraically (a) (A/F, i,n) = (A/P, i, n) - i

Question

Prove the following relationships algebraically (a) (A/F, i,n) = (A/P, i, n) - i (b) (P/F, i, n) = (P/A, i, n) - (P/A, i, n - 1) (c) (P/A, i, n) = (P/F, i, l) + (P/F, i, 2)+...+(P/F, i, n) (d) (F/A, i, n) = [(F/P, i, n) - 1]/i

Explanation / Answer

Kolmogorov definition Given two events A and B with P(B) > 0, the conditional probability of A given B is defined as the quotient of the joint probability of A and B, and the probability of B: This may be visualized as restricting the sample space to B (see Venn diagram). [edit]As an axiom of probability Some authors, such as De Finetti, prefer to introduce conditional probability as an axiom of probability: Although mathematically equivalent, this may be preferred philosophically; under major probability interpretations such as the subjective theory, conditional probability is considered a primitive entity. Further, this "multiplication axiom" introduces a symmetry with the summation axiom for mutually exclusive events:[1] [edit]Definition with ?-algebra If P(B) = 0, then the simple definition of P(A|B) is undefined. However, it is possible to define a conditional probability with respect to a ?-algebra of such events (such as those arising from a continuous random variable). For example, if X and Y are non-degenerate and jointly continuous random variables with density

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