Part A 1. Consider n rolls of a balanced die. Let X, be the outcom...

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Part A 1. Consider n rolls of a balanced die. Let X, be the outcome of the ith roll and let 8.= (a) Find an asymptotic approximation (large n) to the distribution of X (b) Find c, such that 8. 2. Let X14 < X24 < X34 < Xes be the order statistics of a random sample of size 4 from the distribution having pdf f(x) zero elsewhere. Find P(3: < Xes). 3. Let Y denote the nth order statistic of a random sample of size n from a distribution having pdf, f(x) - 1/0, and zero elsewhere. Find the limiting distribution of Xnn 4. Let X, - Gamma(e, N - n), i = 2, n where @ does not depend on n. Find an asymptotic approximation (large n) to the distribution of T. - Xa/n 5. Let of be the sample variance of n independent random variables, ench distributed as 02). Prove that & converges in probability to of Part B 1. Let Y, - xi.) and 2-1-20 - 22 Show that z,Azz Normal(0,1). 2. Let x - Beta(a,6), where a, ,6>0. Derive E(X"). 3. Let X, - Normal(a, and define U - and W - (a) Find a statistics that is a function of U and W that is unbiased for 02 + 12. (b) Let c be a constant, and define Y, - 1 if X, s c and zero otherwise. Find a statistic that is a function of Y1.Y2. Y. and also unbiased for Fx(c) - (+) B11 N (n.) p) BIN(I,p) pri - p) np. np(1-p) (pe'+9)" NB (r.p) CAN GEO(p) prot r/prain? 1/p.g/p² (F) in 1-8 HY P(n,M,N) CON/C) POI(A) cape enter-1) UNI F(a,b) 1/(b-a) N 02) - agb. it 11,02 GAMMA (0,k) EXP(#) no 0,0° (Ita)" Ft EX. P(0,7) DE (0.v) se noto.or i-pe - WEI (0.8) towa, exp - (x/0) EV (0,n) - - @r(1 + 3),ugly e"T(1+01 CAU (0,1)) PAR (0,x) ** x2(v) t (t) (I+E) 22 ** M/2 BETA (a, b) l'(a+b) Theorem (2.4.7 Chebychev]: If X is random variable with mean II and variance 02, then for any * > 0, PIIX- " > ka) s 1/N which can also be written as <ka)>1-1/k2. - </ka)>

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