Introduction to Mathematical Statistics (7th Edition) by Robert V. Hogg, Joeseph McKean, Allen T. Craig

By Robert V. Hogg, Joeseph McKean, Allen T. Craig

Advent to Mathematical facts, 7th version, deals a confirmed method designed to supply you with a superb beginning in mathematical information. plentiful examples and workouts through the textual content illustrate techniques that will help you achieve a pretty good knowing of the fabric.

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The sample space is C = {(i, j) : 1 ≤ i, j ≤ 6}. Because the dice are fair, P [{(i, j)}] = 1/36. The random variable X is X(i, j) = i + j. The space of X is D = {2, . . , 12}. By enumeration, the pmf of X is given by Range value x 2 3 4 5 6 7 8 9 10 11 12 Probability pX (x) 1 36 2 36 3 36 4 36 5 36 6 36 5 36 4 36 3 36 2 36 1 36 To illustrate the computation of probabilities concerning X, suppose B1 = {x : x = 7, 11} and B2 = {x : x = 2, 3, 12}. Then, using the values of pX (x) given in the table, PX (B1 ) = pX (x) = 2 8 6 + = 36 36 36 pX (x) = 2 1 4 1 + + = .

If x ∈ S, then pX (x) is equal to the size of the discontinuity of FX at x. If x ∈ S then P [X = x] = 0 and, hence, FX is continuous at this x. 2. A lot, consisting of 100 fuses, is inspected by the following procedure. Five of these fuses are chosen at random and tested; if all five “blow” at the correct amperage, the lot is accepted. 31931. More generally, let the random variable X be the number of defective fuses among the five that are inspected. The pmf of X is given by pX (x) = 80 (20 x )(5−x) 100 (5) 0 for x = 0, 1, 2, 3, 4, 5 elsewhere.

100, zero elsewhere. (b) Compute P (X ≤ 50). 40 Probability and Distributions (c) Show that the cdf of X is F (x) = [x]([x] + 1)/10100, for 1 ≤ x ≤ 100, where [x] is the greatest integer in x. 10. 1. 11. Let X be a random variable with space D. For D ⊂ D, recall that the probability induced by X is PX (D) = P [{c : X(c) ∈ D}]. Show that PX (D) is a probability by showing the following: (a) PX (D) = 1. (b) PX (D) ≥ 0. (c) For a sequence of sets {Dn } in D, show that {c : X(c) ∈ ∪n Dn } = ∪n {c : X(c) ∈ Dn }.

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