[wiley series in probability and statistics] an introduction to probability and statistics...

Answers to Selected Problems

Problems 1.3

1. (a) Yes; (b) yes; (c) no. 2. (a) Yes; (b) no; (c) no. 6. (a) 0.9; (b) 0.05; (c) 0.95. 7. 1/16. 8. \ + | In 2 = 0.487.

Problems 1.4

«KM9/(?HK,/)-'(H/(?) ,0 [,0(4,= - 4 - 9 ( 4 , ] / ( 5

52 ) ; ( E , . 3 ( ' 2

2 ) ( * ) 4 . / ( 55

2 )

<)G)Q(r)/(?M?)©(?)*/(?)-Problems 1.5

3. a(pby £ ( t 0 [P° ~ b)]l- 4' p/(2 ~ P)-N N , ,

5. ^ 0 7 W + 7 J ] 0 7 W - ~ for large AT. 6. n = 4. y=o 7=0 « + z

10. r/(r + g). 11. (a) 1/4; (b) 1/3. 12. 0.08. 13. (a) 173/480; (b) 108/173,15/173. 14.0.0872.

693

An Introduction to Probability and Statistics, Second Edition by Vijay K. Rohatgi and A. K. Md. Ehsanes Saleh

Copyright © 2001 John Wiley & Sons, Inc.

694 ANSWERS TO SELECTED PROBLEMS

Problems 1.6 1. 1/(2 - p); (1 - p)/(2 - p). 4. p2{\ - />)2[3 - 7/7(1 - />)]. 12. For any two disjoint intervals Iu I2 c (a, b), t{I\)t(I2) = (b - a)t(Ix n 72), where £(/) = length of interval / .

(8/36 ifn = \ 13.(a)p„ = ) 2 ( i r ( | ) 2 + 2 ( i r ( ^ + 2 ( i r ( ^ ) 2 ^ 2 ; (b) 22/45;

(c )12/36;2( i r 2 (^) ( i )+2( i r ( l | ) (A) + 2( i r 2 ( i i ) ( | ) for„=2,3

Problems 2.2 3. Yes; yes. 4.0; {(1,1, 1,1,2), (1, 1, 1, 2, 1), (1, 1, 2, 1, 1), (1,2, 1, 1, 1), (2, 1, 1, 1, 1)}; {(6,6, 6, 6,6)};

{(6,6, 6, 6,6), (6, 6,6,6,5), (6,6,6, 5, 6), (6, 6, 5,6,6), (6, 5, 6, 6, 6), (5, 6, 6,6, 6)}. 5. Yes; (1/4, 1/2) U (3/4, 1).

Problems 2.3

1. X

P(X=x)

0 1 2

1/8 3/8 3/8

3

1/8 F(x) = 0, x < 0, = 1/8, 0 < x < 1; = 1/2, 1 < x < 2; = 5/8, 2 < JC < 3;

= 1, x > 3 . 3. (a) Yes; (b) yes; (c) yes; yes.

Problems 2.4

1. (1 - p)n+l - (1 - p)N+\ N >n.

7T(1 + * 2 ) 3. Yes; F9(x) = 0 x < 0, = 1 - e'** - 0x<r** for x > 0; P(X > 1) = 1 - F0(l).

4. Yes; F(x) = 0, x < 0; = 1 - ( 1 + - ^ — | e~x/e for JC > 0. V 0 + 1/

6. F(JC) = <?72 for x < 0, = 1 - e~x/2 for J C > 0. 8. (c), (d), and (f). 9. Yes; (a) 1/2,0 < x < 1,1/4 for 2 < x < 4; (b) 1/(20), |x| < 0;

(c) xe'x, x > 0; (d) (x - l)/4 for 1 < x < 3, and P(X = 3) - 1/2; (e)2xe-x\x>0.

■10. If S(x) = 1 - F(JC) = P(X > x), then S'(x) = - / ( J C ) .

Problems 2.5

2. X=l/X. 4. 0[1 - exp(-2^0)]yr^[e-^ arc c o s > + *-**-" m c o s >], |y | < 1.

| 0 exp{-0 arctan z}[(l + z2)(l - e~6n]-\ z > 0, } 0 e x p { - 7 r 0 - a r c t a n z } [ ( l + z 2 ) ( l - ^ 7 r ) ] - 1 , z < 0.

10. /wCy) = 2/3 for 0 < y < 1, = 1/3 for 1 < y < 2. 12. (a) 0, y < 0; F(0) for - 1 < y < 1, and 1 for y > 1;

(b) = 0 if y < -b, = F(-b) if y = -ft, = F(y) if -ft < y < b, = 1 if y > ft;

ANSWERS TO SELECTED PROBLEMS 695

(c) = F(y) ify< -by = F(~b) if -b < y < 0, = F(b) if 0 < y < b, = F(y) ify >b.

Problems 3.2

3. EX2r = 0 if 2r < 2m - 1 is an odd integer, r ( m - r + i ) r ( r + i )

= N /«\ / ,\ if 2r < 2m - 1 is an even integer. r ( i ) r ( f i )

9. iP = a( l — v)/v where v = (1 — z?)1^. 10. Binomial: a3 = (q — P)/+Jnp<l> a4 = 3 -f (1 — 6pq)/3npq

Poisson: ft3 = Arl/2, a4 = 3 -f 1/A..

Problems 3.3

1. (b) e~HeXs - 1)/(1 - «-*); (c) p[l - (qs)N+l]/[(l - qs)(\ - qN+{)l s < 1/q. 6. f(6s)/f(0\ f{e*)lf{6).

Problems 3.4 a2 / a 2 \ x2

3. For any a2 > 0 take P(X = x) = — -, P (X = = — - , x ^ 0. a 2 + JC2 V x ) a2 -{-x2

( a4K2-n4\ a4[K2 - l ] 2 ^

P(X2 = K2a2) = M4 a ' fi4 + AT4a4 - 2t f 2a4 '

Problems 4.2

1. No. 4. 1/6; 0. 7. Marginals negative binomial, so also conditionals. 8. h(y\x) = \{c2 + JC2)/(C2 + x2 + j 2 ) ^ 2 . 9. X - B(p<, p2 + pa); K/(l - x) - #(/>2, p3). 10. X - G(a, 1/0), F - G(<* + y, 1/0), X/y - B(a, y), K - x - G(y, l/fi). 14. P(X < 7) = 1 - e~\ 15. 1/24; 15/16. 17. 1/6.

Problems 4.3 3. No, yes, no. 10. = 1 - a I(2b) if a < b, = b/(2a) ifa>b. U.k/{k + fi)9l/2.

Problems 4.4

2. (b) fV]U(v\u) = l/(2w), |v| < **, w > 0. 6. P(X = JC, M = m) = TT(1 - jr)m[l - (1 - 7r)'"+1] if x = m, = TT2(1 - w)

if x < m. P(M = ro) = 2JT(1 - 7r)m - JT(2 - TT)(1 - TT)2"1, m > 0.

7. /X(JC) = \ke~x/k\, k < JC < A: + 1, A: = 0,1,2,.. . . 11. / „ < » = 3 M 2 / ( 1 + M ) 4 , M > 0 .

i/n+jt


13. (a) FUtV(u, v) = [i - exp (~^J ] ( ^ ^ ) if u > °- N ^ */2>

= 1 - exp[l - u2/(2a2)] if u > 0, u > 7r/2, = 0 elsewhere; 1 2U'/2-!e-«>/2

(b) / ( l l , V) = -=^""" 7=r.

V^ ro/2)V2

Problems 4.5

2e+] 2€+2

2. EXkYl = - — — - — - + ———-——. 3. cov(X, Y) = 0; X, K dependent. (ft + 3)(£ + l) 3(it + 2)(€ + 2) ^

15. Afy,v(ii, v) = (1 - 2v)~l exp{a2/(l - 2v)} for v < 1/2; p(U, V) = 0; no. 18. pz,w = (^2 ~ i2) sin 0 cos 6>/Vvar(Z) var(W).

21. If U has pdf / , then EXm = EUm/(m + 1) for m > 0; p = \ - -= = . F " 2 §var(t/) + f(E£/)2

Problems 4.6

1. M + a [ / ( ? ^ \ - f ( t l £ \ ]/d> ( ^ ) - * ( ^ ^ ) ] where <D is the standard

normal df. 2. (a) 2(1 + X). 3. E{X\y) = m + p^ty - /*2). 4. £(var{r|X}). 6.4/9. 7.(a)l; (b) 1/4. 8. xk/(k + 1), 1/(1 +k)2.

Problems 4.7

5 « ( l > j W (b) r.

Problems 5.2

'•«3r)-(i)/U>'<'-,)-(i-.1I)/(2).,E« + l.-

W _ A » ) . ! / ( ; ) ; FO , . | r = ) , ) = ^ 5 ^ L . 0 < » , < , , i = 1,... , j , x,- x; for / / 7.

9. P{YX = x)=qpx +pq\x> 1; P(F2 = *) = P V ' 1 -f-^V"1 '* > 1; P(Fn = X) = P(F, = x) for rc odd; = P(Y2 = x) for n even.

Problems 5.3

2. (a) P JF(X) = ELo ( j ) P*d - P)"-*| = ( " ) Pxd ~ P)n-\x = 0, 1,... , n.

22. X/\Y\ ~ C(l, 0); (2/JT)(1 + z2y\ 0 < z < oo. 27. (a) f/a2; (c) = 0 if t < 0, = a/t if t > 0; (d) (a/P)ta-]. 29. (b) 1/(2VJT), 1/2.


Problems 5.4

1. (a) MI =4 , /x 2 = 15/4, p = - 3 / 4 ; ( 5 ) ^ ( 6 - £ x , g ) ; (c) 0.3191. 4. &A/*(a/z, + b, cii2 + </, 02of, c2a2, p). 6. tan2 0 = EX2/EY2. 7. a2 :

Problems 6.2

l.No. 2. Yes. 3. Yn -> y - F(y) = Oif y < 0, = 1 - * r ^ if y > 0. 4. F(y) = Oif y < 0, = 1 - e~y if y > 0. 9.C(1,0). 12. No. 13. (a) exp(-jc-a), J C > 0; EXk = T(l - */«), k < a;

(b) exp(-*r*), - o o < x < oo; M(f) = T(l -t),t < 1; (c) exp{-(-Jt)ff}, JC < 0; FX* = ( - l ) * r ( l + k/a), k > -a.

20. (a) Yes, no; (b) yes, no.

Problems 6.3

3. Yes; An = n(n + 1 )M/2 , fl„ = a V*(« + l)(2n + l)/6.

5. (a) Mn(t) -> 0 as n -> oo, no; (b) M„(f) diverges as n ^ oo; (c) yes; (d) yes; (e) Mn -» e' /4, no.

Problems 6.4

1. (a) No; (b) no. 2. No. 3. For a < 1/2. 7. (a) Yes; (b) no.

Problems 6.5

4. Degenerate at p. 5. Degenerate at 0.

6. For p > 0, JV(0 , ^/p), and for p < 0, Sn/n —► degenerate.

Problems 6.6

1. (b) No; (c) yes; (d) no. 2. JV(0, 1). 3. Af(0, cr2/p2). 4. 163 8. 0.0926; 1.92.

Problems 7.2

1. PX = 0) = P(X = 1) = 1/8, P(X = 1/3) = P(X = 2/3) = 3/8, P(S2 = 0) = 1/4, P(S2 = 1/3) = 3/4.

X

/>(*)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

Problems 13

l . {F(nun(*,y)) -F(x)F(y)} /» .


6, E(S2)k = j£vp(n - l)(n 4- 2) . . . (« + 2* - 3), it > 1.

9. (a) P(X = /) = e-"x(«A.)m/0«)K / = 0, l /n, 2 / / i , . . . ; (b) C(1,0); (c) r(#im/2, 2/n). 10. (b) 2/V««; 3 + 6/(a/i).

11. 0,1,0, E(Xn - 0.5)7(144n2). 12. var(.S2) = ± (X + ^ - ) > var(X)

Problems 7.4

2. n(m + 5)/[m(n - 2)]; 2n2{(m + <5)2 + (/i - 2)(m'+ 25)}/[m2(« - 2)2(n - 4)].

3. W» l W t n > 1; » (1 + _ L /*A 2 J © 2 r ( ^ \ « - 2 x g , / ? / » x . « > 2 .

(I) P - ,2

11. 2mm/2nn/2(n + meu)-(fn+n)riea*IB ( ~ , - ) , - o o < z < oo.

Problems 7,5

1. (a) AN(/x2,4^V2) for fi £ 0, X2/a2 - ^ x2( l ) for /x = 0, a 2 = a2/n;

(b) for xx # 0,1/X ~ AN(l//x, a2//x4); for xx = 0, a„/X„ - ^ l/A/"(0,1);

(c) for /x ^ 0 , In |X| - AN(ln |/x|, a2/xx2); for /x = 0, ln(|X|/a„) -^> In |JV(0, l ) | (d) AN(e<\ e2<V2).

2. c = 1/2 and Vx ~ AN(Vx, 1/4).

Problems 7.6

,.,<„-„. , „» + ,-2>. , ( i i L ) ' r ( ^ + t)/r(^l).

Problems 7.7 r 2 . 2 i -,-(0/2+1)

3. t2,(i - ^)i->/2 n + yl±Azl^yi\ , b o t h „ ,(M). L «(i-p2) J

4. Vn^Tr~/(n- 1).

Problems 8.3

T.A(x)/A,d). 9. No. 10. No.

ll.(b)X(n); (e)(X,52); (g) f f jx, , fj(l - X,)J; (h) X((l), Xm X(n))

Problems 8.4

o (n±Y r ( " 2 ) „. f n i V2 r V + 2 ) V 2 ) r/n + p-iy '\ 2 ) r / » + 2 p - l \


3. St? = ^ S \ var(5?) = ( ^ } ) 2 & < var(52) = £ . 4. No. 5. No.

6 - ( a ) ( ; i 5 * ) / ( " ) . 0 < * < * < » . / = E : ^ (b)= Q / ( " ) i f 0 < / < 5 ;

- 2 / ( ; ) i f r - , . » d ( ; : ; ) / ( ; ) i f , + i < r < . .

9 C + " " 2 ) / C + " _ 1 ) ' ' = E j c / n(a)/vx/": (b)n°-12. t = HlXi, 1 - (1 - * ) " - ' iff > t0, and 1 iff < t0.

n.(a)mtht = Y,"xj'T.UnnJ"' (b)4)i ' '"^s ; (c) a - i/«)'; ( d ) ( l - l / n ) ' - ' [ l + t l ] . .

14. With / = *<„), [tnf(t) - ( t - l)nf(t - l)]/[/" - (< - 1)"], r > 1.

15.WW,» = n x y . ( j ) ( l ) 4 ( l - ^ .

Problems 8.5

1. (a), (c), (d) Yes; (b) no. 2. 0.64761/w2. 3. n~l sup{jc2/[^2 - 1]}. 5. 2(9(1 - 0)/n.

xj-0

Problems 8.6

2.J3 = (n~ l)S2/(wX), a = X//3. 3. £ = X, a2 = (n - l)S2/n.

4. a = X(X - X^fX2 - X 2 ] - J , X2" = £ ? *?/» 0 = (1 - X)(X - X ^ X 2 - X2]-1 .

5. ft = lnfx'/fX*]1/2}, £ 2 = ln{XVX2}, X2 = E i *,2/"-

Problems 8.7

l.(a)med(X,); (b) X(1); ^c) n/ £ ? * £ ( d ) - " / £ > ( 1 ~ * , ) • __ 2. (a) X/n; (b) ft, == 1/2 if X < 1/2, = X if 1/2 < X < 3/4, = 3/4 if X > 3/4;

^ = - ! - x / ^ + (f)2^=E^2/«; (d) (9 = ^ j if «i, «3 > 0; = any value in (0,1) if nx =n3 = 0; no mle ifn\ = 0, n3 / 0; no mle if W] ^ 0, w3 = 0;

(e) § = - I + 1VT+4X 2 ; (f) 0 = X. 3. /z = - ^ - ' ( m / t t ) . 4. (a)fi = X(1), 0 = £ ? ( * , ~ «) /» ; (b) A = ft./iCXi > 1) = e*-1* c* < 1,

= 1, <*_> 1; A = 1 if a > 1, = exp{(a - l)/0] if a < 1. 5. 0 = 1/X. 6. /t = E In X,/«, a2 = ^ ( l n X , - jx)2/n. 8.(a)N = ^ i X ( M ) - l ; (b)X(M). 9- Ai = £"=, Xu/n = * „ * = 1, 2 , . . . , j , a2 = ES(Xl7 - X,)2/(ns). 11. A = ? • 13. </(#) = (X/w)2. 15. /t = max(X, 0). 16. pj = Xj/nJ = 1,2, . . . ,k- 1.


Problems 8.8

2. (a) (Ex,-+ l)/(« + 1); (b) ( ^ ) E x / + 1 . 3. X. 5. X/n. 6. (X + 1)(X + «)/[(« + 2)(n + 3)]. 8. (a + n) max(a, X(n))/(a + « - 1).

Problems 8.9

5. (c) (n + 2)[(X(ll)/2))-<"+,> - (X(,))-<"+I>]/{(/i + l)[(X(n)/2)-("+2> - (*(I))-<"+2>]}. 10.(E*I-)*r(rt + * ) / r (n + 2*).

Problems 9.2

1. 0.019, 0.857. 2. * = /x0 + tf W v ^ , 1 - * (z„ - ^ f ^ v ^ ) -5. exp(-2), exp(-2/0), 6 > 1.

Problems 9.3

1. 0(x) = 1 if x < 0O(1 — \ / l - a) , = 0 otherwise. 4. 0(x) = 1 if ||*| - 1| > k. 5. <p(x) = 1 if JC(1> >c = 90- ln(a1/n). ll.Ifflo < 0x,4>(x) = 1 if x(1) > 0oa"1/n,andif<9, < 0o,then0(x) = 1 ifx(1) < 6 > o ( l - a , / n ) - , . 12. 0(JC) = 1 if x < Jail or > 1 - Ja/2.

Problems 9.4

1. (a), (b), (c), and (d) have MLR in EX,; (e) and (0 in fli *>• 4. Yes. 5. Yes, yes.

Problems 9.5 1. <t>(xi,x2) — 1 if |JCI — x2\ > c, = 0 otherwise, c = y/2za/2-2. 0(x) = 1 if Ex, > k. Choose k from a = P^ ( £ " X, > Jt).

Problems 9.6

3. 0(x) = 1 if (no. of JC/'S > 0 - no. of x,-'s < 0) > k.

Problems 10.2

2. Y = # of x\, JC2 in sample, 7 < ct or F > c2. 3. X < Ci or > c2. 4. S2 > c\ or < c2. 5. (a) X(n) > N0; (b) X(n) > N0or < c. 6. |X - 6b/2| > c. 7. (a) X < d or > c2; (b) X > c. 11. X(i) > 00 - ln(a)1 /" . 12. X(1) > 0oa", /n .

Problems 10.3

1. Reject at a — 0.05. 3. Do not reject //0 : p\ = p2 = /?3 = P4 at 0.05 level.


4. Reject H0 at a = 0.05. 5. Reject at 0.10 but not at 0.05 level. 7. Do not reject H0 at a = 0.05. 8. Do not reject H0 at a = 0.05. 10. £/ = 15.41. 12. P-value = 0.5447.

Problems 10.4

1. t = -4 .3 , reject //0 at a = 0.02. 2. f = 1.64, do not reject //0. 5. t = 5.05. 6. Reject f/0 at a = 0.05. 7. Reject #0- 8. Reject //0.

Problems 10.5

1. Do not reject H0 : <*\ = <T2 at a = 0.10. 3. Do not reject //0 at or = 0.05. 4. Do not reject H0.

Problems 10.6

2. (a) <£(x) = 1 if Ex, = 5, = 0.12 if Ex, = 4, = 0 otherwise; (b) minimax rule rejects H0 if EJC, = 4 or 5, and with probability 1/16 if E ^ = 3; (c) Bayes rule rejects H0 if EJC, > 2.

3. Reject H0 if* < (1 - l / / i) ln2; jg(l) = P(Y < (n - l) ln2), /3(2) = P(Z < (w - l)ln2) where Y - G(n, 1), and Z - G(w, 1/2).

Problems 11.3

1. (77.7,84.7). 2. n = 42. 7. f-2^, 2£X</x22„,!_a/2Y

9. (2X/(2 - A.0, 2X/(2 - X2)), A2 - A2 = 4(1 - a) . 10. [a1 '" / / ] .

X 1 " - [Ind+rf/^))!' __

12. Choose it from a = (A; + 1)*-*. 13. X + zao/Jn. 14. (EX?/c2, EX2/c,) where / ^ x'Cytfy = 1 - a, and £ 2 yx*200</y = » 0 - «). 15. Posterior #(w + a, Ex, -f 0 - «). 16. A(A*W = y ^ e x p l - f (M - J)2}[d>(v^(l - x)) - 4>(-V*0 + i ) ) ] , where <D

is standard normal df.

Problems 11.4

l . (X ( ! i -x 22

a / (2«) ,X ( 1 ) ) .

2. (2HX/&, 2nX/a\ choose A, & from / j 7 xlWdu = 1 - or, and fi2x£(a) = b2xln(b), where x 2 W *s t n e pdf of x2(*0 rv.

3. (X/(l - b), X/( l - a)), choose a, £> from 1 - a = fc2 - a2 and a(l - a)2 = £>(1 - b)2, 4. /i = [4z2_a/2/t/2] + 1; n > (1/a) ln(l /a).

Problems 11.5

\.(Xin),a-l"XM). 2. (2EX,/X2, 2EX./X,) where A.|, A2 are solutions of A,/2«a(Ai) = ^2/2*0(^2) and


P(l) = l - a , / B i s x 2 ( t ; ) p d f .

3. (X(1) - | f , X(l)). 5. («1/nX(]), X(1)). 8. Yes.

Problems 12.3

4. Reject H0 : a0 = a'0 if , Q V ^ > c0.

8. Normal equations ftSjcf + ft Ssf+I + ftX!jcf+2 = S^ocf, it = 0,1,2.

Reject Ho : ft = 0 if [\h\lyf^M^(Xi ~ ft - ft*; - A*?)} > *b where

ft = Ec/K,- and ft - 7 - ftl, ft = EC*, - 3 0 ^ - 7 ) / E t a - Y)1. 10. (a) ft = 0.28, ft = 0.411; (b) t = 4.41, reject //0.

Problems 12.4

2. F = 10.8. 3. Reject at a = 0.05 but not at a = 0.01. 4. BSS = 28.57, WSS = 26, reject at a = 0.05 but not at 0.01. 5. F = 56.45. 6. F = 0.87.

Problems 12.5

4. SS methods = 50, SS ability = 64.56, ESS = 25.44; reject H0 at a = 0.05, not at 0.01. 5. Variety = 24.00.

Problems 12.6

2. Reject H0 if ^ ^ ^ Z J' - ,2 > c. fl«£i(y.ry)

EEE(y / y s -y f , , )2

4. SS! (machines) = 2.786, d.f. = 3; SSI = 73,476, d.f. = 6; SS2 (machines) = 27.054, d.f. = 2; SSE = 41.333, d.f. = 24.

5. Cities 3 227.27 4.22 Auto 3 3695.94 68.66 Interactions 9 9.28 0.06 Error 16 287.08

Problems 13.2

l.dis estimable of degree 1; (number of jt/'s in A)/n. 2. (a) (mn)'1 EX, E Y}; (b) S\ + Si _ 3. (a) HXiYi/n\ (b) E(Xt + Yx- X - Y)2/(n - 1).

Problems 13.3

3. Do not reject H0. 7. Reject H0. 10. Do not reject H0 at 0.05 level. 11. T+ = 133, do not reject #0 . 12. (2nd part) T+ = 9, do not reject H0 at a = 0.05.


Problems 13,4

1. Do not reject H0. 2. (a) Reject; (b) reject. 3. U = 29, reject //0. 5. d - \, do not reject H0. 7. / = 313.5, z = 3.73, reject; r = 10 or 12, do not reject at a = 0.05.

Problems 13.5

1. Reject H0 at a = 0.05. 4. Do not reject //0 at a = 0.05. 9. (a) t = 1.21; (b) r = 0.62; (c) reject H0 in each case.

Problems 13.6

1. (a) 5; (b)8. 3. pn~2(n + p - np) < 1. 4. M > (Zi_yVP0(l ~Po) ~ Z\-ty/p\(\- P\))2I(P\ ~ Po)2.

Problems 13.7

1. (c) E{n(X - IJL)2}/ES2 = 1 + 2p(l - 2p/n)~]; ratio = 1 if p = 0, > 1 for p > 0. 2. Chi-square test based on (c) is not robust for departures from normality.

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