Download Schaum's Outline of Probability and Statistics, 3rd Ed. by John Schiller, R. Alu Srinivasan, Murray Spiegel PDF

By John Schiller, R. Alu Srinivasan, Murray Spiegel

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97. (a) A shelf contains 6 separate compartments. In how many ways can 4 indistinguishable marbles be placed in the compartments? (b) Work the problem if there are n compartments and r marbles. This type of problem arises in physics in connection with Bose-Einstein statistics. 98. (a) A shelf contains 6 separate compartments. In how many ways can 12 indistinguishable marbles be placed in the compartments so that no compartment is empty? (b) Work the problem if there are n compartments and r marbles where r Ͼ n.

A) A shelf contains 6 separate compartments. In how many ways can 12 indistinguishable marbles be placed in the compartments so that no compartment is empty? (b) Work the problem if there are n compartments and r marbles where r Ͼ n. This type of problem arises in physics in connection with Fermi-Dirac statistics. 99. A poker player has cards 2, 3, 4, 6, 8. He wishes to discard the 8 and replace it by another card which he hopes will be a 5 (in which case he gets an “inside straight”). What is the probability that he will succeed assuming that the other three players together have (a) one 5, (b) two 5s, (c) three 5s, (d) no 5?

15. Prove Theorem 2-1, page 42. The probability function for U is given by g(u) ϭ P(U ϭ u) ϭ P[f(X) ϭ u] ϭ P[X ϭ c(u)] ϭ f [c(u)] In a similar manner Theorem 2-2, page 42, can be proved. 16. Prove Theorem 2-3, page 42. , u increases as x increases (Fig. 2-11). There, as is clear from the figure, we have (1) P(u1 Ͻ U Ͻ u2) ϭ P(x1 Ͻ X Ͻ x2) or (2) u2 x2 1 1 3u g(u) du ϭ 3x f (x) dx Fig. , the slope is positive). 67). The theorem can also be proved if cr(u) Ն 0 or cr(u) Ͻ 0. 17. Prove Theorem 2-4, page 42.

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