By Fernandes M.

This compendium goals at offering a finished assessment of the most themes that seem in any well-structured path series in statistics for enterprise and economics on the undergraduate and MBA degrees.

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**Statistics for Business and Economics**

This compendium goals at delivering a entire review of the most subject matters that seem in any well-structured path series in information for company and economics on the undergraduate and MBA degrees.

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It suﬃces to compute the expectation of H(X) given the density function of X. com 64 Statistics for Business and Economics Example: Probability distributions In some situations, the interest lies on the magnitude of the random variable regardless of the sign it takes. Suppose, for instance, that X has a double exponential density given by fX (x) = ⎧ ⎨ 1 ex if x ≤ 0 2 ⎩ 1 e−x if x ≥ 0, 2 which is symmetric around zero. Now, the expected value of Y = |X| is E(Y ) = = ∞ −∞ 1 2 |x| fX (x) dx = 0 0 1 2 −∞ ∞ 0 ∞ xe−x dx = (−x)ex dx + −∞ ∞ |x| ex dx + 0 |x| ex dx xe−x dx = 1, 0 where the last equality follows from integration by parts.

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We employ quotation marks because integrating out a discrete random variable, say Y , corresponds to summing the joint probability for all possible values of Y given that any of these values may occur. Letting Bj = {X = x, Y = yj } for j = 1, 2, . . then ∞ j=1 yields p(x) = Pr(X = x) = Pr(B1 or · · · or Bn or · · · ) = p(x, yj ), given that these events are all mutually exclusive. As for continuous random variables, the marginal density function of X is fX (x) ∼ Pr(x ≤ X ≤ x + Δx) for a very small Δx > 0 ∞ = Pr(x ≤ X ≤ x + Δx, −∞ < Y < ∞) = −∞ fXY (x, y) dy, and hence b Pr(a < X < b) = Pr(a < X < b, −∞ < Y < ∞) = a ∞ −∞ fXY (x, y) dy dx.