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Final Exam Study Guide - Statistical Theory I | ST 521, Study notes of Statistics

North Carolina State University (NCSU)Statistics
Prof. Sujit Ghosh

Final Exam Material Type: Notes; Professor: Ghosh; Class: Statistical Theory I; Subject: Statistics; University: North Carolina State University; Term: Unknown 1989;

Typology: Study notes

Pre 2010

Uploaded on 03/18/2009

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Practice Exercises for Final Exam
ST 521 - Statistical Theory - I
The ACTUAL exam will consists of less number of problems.
1. Suppose X1, X2, . . . , Xn
iid
∼fX(x), where fX(x) is the pdf of a random variable X.
(a) Show that U=FX(X)∼U(0,1), where FX(x) = Rx
−∞ fX(t)dt.
(b) Define Ui=FX(Xi) for i= 1,2, . . . , n. Show that U1, U2, . . . , Un
iid
∼U(0,1).
(c) Let X(1), X(2), . . . , X(n)denote the order statistics corresponding to the ran-
dom sample of size nfrom fX(x). Define U(i)=FX(X(i)) for i= 1,2, . . . , n.
Show that U(1), . . . , Un)are the order statistics corresponding to U1, . . . , Un.
(d) Show that FU(n)(u) = Pr[U(n)≤u] = unif 0 <u<1 and hence show that
U(n)∼Beta(n, 1). Obtain the mean and variance of U(n).
(e) Use (c) and (d) to obtain the cdf and pdf of X(n).
Show that E[X(n)] = nR1
0F−1
X(u)un−1du, where F−1
X(u) = inf{x:FX(x)≥u}
2. Suppose Xi
indep
∼N(i−3, i2) for i= 1,...,5. Use Xi’s to construct a statistic with
the indicated distribution.
(a) a N(0,1) distribution,
(b) a χ2
5distribution,
(c) a t5distribution,
(d) a F1,5distribution,
(e) a Gamma(2,3) distribution,
(f) a U(0,1) distribution,
(g) a Beta(1,2) distribution,
(h) a DE(0,1) distribution,
(i) a Geometric(0.75) distribution,
(j) a NB(5,0.25) distribution, and
(k) a F3,2distribution.
ST 521: Practice Problems - II Page 1 c
Sujit Ghosh, NCSU Statistics
pf3

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Practice Exercises for Final Exam

ST 521 - Statistical Theory - I

The ACTUAL exam will consists of less number of problems.

  1. Suppose X 1 , X 2 ,... , Xniid ∼ fX (x), where fX (x) is the pdf of a random variable X.

(a) Show that U = FX (X) ∼ U (0, 1), where FX (x) = ∫^ −∞x fX (t)dt. (b) Define Ui = FX (Xi) for i = 1, 2 ,... , n. Show that U 1 , U 2 ,... , Uniid ∼ U (0, 1). (c) Let X(1), X(2),... , X(n) denote the order statistics corresponding to the ran- dom sample of size n from fX (x). Define U(i) = FX (X(i)) for i = 1, 2 ,... , n. Show that U(1),... , Un) are the order statistics corresponding to U 1 ,... , Un. (d) Show that FU(n) (u) = Pr[U(n) ≤ u] = un^ if 0 < u < 1 and hence show that U(n) ∼ Beta(n, 1). Obtain the mean and variance of U(n). (e) Use (c) and (d) to obtain the cdf and pdf of X(n). Show that E[X(n)] = n ∫^01 F (^) X− 1 (u)un−^1 du, where F (^) X− 1 (u) = inf{x : FX (x) ≥ u}

  1. Suppose Xiindep ∼ N (i − 3 , i^2 ) for i = 1,... , 5. Use Xi’s to construct a statistic with the indicated distribution. (a) a N (0, 1) distribution, (b) a χ^25 distribution, (c) a t 5 distribution, (d) a F 1 , 5 distribution, (e) a Gamma(2, 3) distribution, (f) a U (0, 1) distribution, (g) a Beta(1, 2) distribution, (h) a DE(0, 1) distribution, (i) a Geometric(0.75) distribution, (j) a N B(5, 0 .25) distribution, and (k) a F 3 , 2 distribution.
  1. Let (X, Y ) be a pair of ransom variables with finite variances.

(a) Show that ˆg(x) = E[Y |X = x] minimizes E[Y −g(X)]^2 over all functions g(x) such that E[g^2 (X)] < ∞. (b) Let  = Y − gˆ(X). Show that i. E[] = 0 and V ar[] = E[V ar[Y |X]]; ii. Cov[X, ] = 0; and more generally, iii.  is uncorrelated with any function g(X) such that E[g^2 (X)] < ∞. (c) Show that V ar[Y ] = V ar[] + V ar[ˆg(X)]. (in above ˆg(x) is known as the least-squares regression function)

  1. Suppose X and Y are independent random variables with finite variances.

(a) Show that Corr[X, Y ] = 0. (b) Show that Corr[|X|,

√ |Y |] = 0. (c) Show that Corr[2X − 3 Y, 3 X − 2 Y ] > 0. (d) Show that Corr[2X + 3Y, 3 X − 2 Y ] = 0 if V ar[X] = V ar[Y ]. (e) Show that Corr[g(X), h(Y )] = 0, provided g(X) and h(Y ) have finite vari- ances.

  1. Suppose X and Y are uncorrelated random variables with finite variances.

(a) Are X and Y necessarily independent? (b) If the joint distribution of (X, Y ) is a bivariate normal distribution, show that X and Y are independent. (c) Given an example of a pair of uncorrelated random variables that are not independent. (d) Suppose U 1 = a 1 + a 2 X and U 2 = b 1 + b 2 Y. Show that Corr(U 1 , U 2 ) = 0. Find a 1 , a 2 , b 1 , b 2 (in terms of means and variances of X and Y ) such that E[U 1 ] = E[U 2 ] = 0 and V ar[U 1 ] = V ar[U 2 ] = 1. (e) Show that c 1 U 1 + c 2 U 2 is positively correlated with U 1 iff c 1 > 0.