Leave X alone a r.v. with p.d.f. f given by:

Iclx-, 1 < x < 0 f (x) = c2(I x2). 0 < x < I 0. something else.

(I) If it is additionally given that EX = 0, decide the constants c1 and c2.

(ii) Determine the ;4-guantile of the appropriation. Clue. To some extent (I), two relations are required for the assurance of al, C2.

14..

Leave X alone a constant r.v. with pth quantile xp, and let Y= g(X"): where g is a rigorously expanding capacity. so the backwards g-1 exists (and is likewise rigorously expanding). Let yp be the pth quantile of the r.v. Y. (I) Show that the pth quantile of Y is given by g(xp): i.e., yp = g(xp).

(ii) If the p.d.f. of X is f (x)= a X, x >0, decide x0 as far as p.

(iii) Find the relating yp of part (I), if Y = ex.

15..

Leave X alone a r.v. disseminated as 8(n, p), and review that P(X= x)= f (x)= Opxqn-x, x = 0, 1, , 11 (q =1 - p). Set B(n, p; x)= f (x).

(I) By utilizing the relationship (13; = ry, , show that:

B(n + 1, p; x)= pB(n, p; x - 1)- 1-qB(n, p; x).

(ii) By utilizing this recursive connection of B(n + 1, p; .), figure the probabilities B(n, p; x)for n = 26, p= 0.25, and x= 10.

16..

Assume that 15 individuals, picked indiscriminately from a (huge) target populace: are inquired as to whether they

favor a specific proposition. It 43.75% ot the objective populace favor the proposition, figure the

likelihood that:

(I) At least 5 of the 15 surveyed favor the proposition.

(ii) A dominant part of those surveyed favor the proposition.

(iii) Compute the normal number ofthose preferring the proposition, and the s.d. around this

number.