By Fernandes M.

This compendium goals at offering a entire evaluate of the most subject matters that seem in any well-structured direction series in facts for enterprise and economics on the undergraduate and MBA degrees.

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**Additional info for Statistics for Business and Economics**

**Example text**

X1 < x2 < . }, then Pr(X = xn ) = Pr(X ≤ xn ) − Pr(X ≤ xn−1 ) = FX (xn ) − FX (xn−1 ). Example: Let X ∈ {x1 , x2 , x3 } with p(x1 ) = 13 , p(x2 ) = 16 , and p(x3 ) = 12 . The cumulative distribution function then reads ⎧ ⎪ ⎪ ⎪ 0, if −∞ < x < x1 ⎪ ⎪ ⎪ ⎨ 1/3 if x ≤ x < x 1 2 FX (x) = ⎪ 1/2 if x ≤ x < x ⎪ 2 3 ⎪ ⎪ ⎪ ⎪ ⎩1 if x3 ≤ x < ∞. 3 Probability distributions Continuous random variable We say that a random variable is continuous if the probability of observing any particular value in the real line is zero.

That is indeed the case because ∞ ∞ (1 − α)α = (1 − α) Pr(X ≥ t) = r r=t = (1 − α) αr r=t t α = αt , 1−α where the penultimate equality comes from the fact that the inﬁnite sum of a geometric progression is equal to the ﬁrst term of the progression divided by one minus the quotient of the progression. 2 Probability distributions Cumulative probability distribution function The goal not always lies on computing a pointwise probability. , the probability of observing X ≤ x. This motivates us to deﬁne the cumulative distribution function as FX (x) = Pr(X ≤ x) = p(xj ), ∀xj ≤ x.

Example: Let X ∈ {x1 , x2 , x3 } with p(x1 ) = 13 , p(x2 ) = 16 , and p(x3 ) = 12 . The cumulative distribution function then reads ⎧ ⎪ ⎪ ⎪ 0, if −∞ < x < x1 ⎪ ⎪ ⎪ ⎨ 1/3 if x ≤ x < x 1 2 FX (x) = ⎪ 1/2 if x ≤ x < x ⎪ 2 3 ⎪ ⎪ ⎪ ⎪ ⎩1 if x3 ≤ x < ∞. 3 Probability distributions Continuous random variable We say that a random variable is continuous if the probability of observing any particular value in the real line is zero. Accordingly, the notion of probability distribution function is meaningless and we have to come up with something a bit diﬀerent, though with a similar interpretation.