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information. 1.98. (a) A shelf comprises 6 separate booths. In what number methods can 12 indistinguishable marbles be positioned within the booths in order that no compartment is empty? (b) paintings the matter if there are n cubicles and r marbles the place r Ͼ n. this kind of challenge arises in physics in reference to Fermi-Dirac facts. 1.99. A poker participant has playing cards 2, three, four, 6, eight. He needs to discard the eight and exchange it by means of one other card which he hopes can be a five (in which case he will get an “inside.

instance 2.6 (a) locate the distribution functionality for the random variable of instance 2.5. (b) Use the results of (a) to discover P(1 Ͻ x Յ 2). (a) we have now x F(x) ϭ P(X Յ x) ϭ three f (u) du Ϫ` If x Ͻ zero, then F(x) ϭ zero. If zero Յ x Ͻ three, then x x 1 x3 F(x) ϭ three f (u) du ϭ three u2 du ϭ 27 zero zero nine If x Ն three, then three x three x 1 F(x) ϭ three f (u) du ϩ three f (u) du ϭ three u2 du ϩ three zero du ϭ 1 zero three zero nine three hence the necessary distribution functionality is zero xϽ0 F(x) ϭ • x3 >27 zero Յ x Ͻ three 1 xՆ3 observe that F(x) raises monotonically from zero to at least one as.

1 is indicated within the decrease right-hand nook of the desk. The joint distribution functionality of X and Y is outlined by way of F(x, y) ϭ P(X Յ x, Y Յ y) ϭ a a f (u, v) (19) uՅ x vՅ y In desk 2-3, F(x, y) is the sum of all entries for which xj Յ x and yk Յ y. 2. non-stop CASE. The case the place either variables are non-stop is acquired simply through analogy with the discrete case on changing sums by way of integrals. hence the joint likelihood functionality for the random variables X and Y (or, because it is in most cases.

P(A2 u A1). (a) for the reason that for the 1st drawing there are four aces in fifty two playing cards, P(A1) ϭ four > fifty two. additionally, if the cardboard is changed for the second one drawing, then P(A2 u A1) ϭ four > fifty two, seeing that there also are four aces out of fifty two playing cards for the second one drawing. Then four 1 four ≤¢ ≤ ϭ fifty two fifty two 169 P(A1 > A2) ϭ P(A1) P(A2 u A1) ϭ ¢ (b) As partly (a), P(A1) ϭ four > fifty two. in spite of the fact that, if an ace happens at the first drawing, there'll be merely three aces left within the last fifty one playing cards, in order that P(A2 u A1) ϭ three > fifty one. Then P(A1 > A2) ϭ P(A1) P(A2.

(c) one is white and one is black. permit W1 ϭ occasion “white ball from first bag,” W2 ϭ occasion “white ball from moment bag.” (a) P(W1 > W2) ϭ P(W1) P(W2 u W1) ϭ P(W1) P(W2) ϭ ¢ three 1 four ≤¢ ≤ ϭ 4ϩ2 3ϩ5 four (b) P(Wr1 > Wr2 ) ϭ P(Wr1 ) P(Wr2 u Wr1 ) ϭ P(Wr1 ) P(Wr2 ) ϭ ¢ five five 2 ≤¢ ≤ ϭ 4ϩ2 3ϩ5 24 (c) the mandatory likelihood is 1 Ϫ P(W1 > W2) Ϫ P(Wr1 > Wr2 ) ϭ 1 Ϫ five thirteen 1 Ϫ ϭ four 24 24 1.14. end up Theorem 1-10, web page 7. We end up the concept for the case n ϭ 2. Extensions to bigger values of n are simply made.