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Exercise 7 (Ancilliarity) Choose one: 1. Let {X;} –1 be independent and identically distributed observations from...
4. Let Xi i = 1,2, ... be independent and identically distributed (i.i.d.) Possion. That is...
4. Let Xi i = 1,2, ... be independent and identically distributed (i.i.d.) Possion. That is let Px;(k) = Px(k) = 2*24. Let Zn = ??=1 X; and find Pzn. Hint: use characteristic functions. n • Use Li=
Observations X1,..., Xn are independent identically distributed, following the PDF fx:(xi) = 0x8-1, and that 0<Xi...
Observations X1,..., Xn are independent identically distributed, following the PDF fx:(xi) = 0x8-1, and that 0<Xi <1 for all i. The parameter is an unknown positive number. Find the ML estimator of e
6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random varia...
6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when
6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed,...
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed, all with the exponential distribution of parameter 1 = 1). a) Compute the cdf of Yn = min(X1,...,xn). b) How do P({Y, St}) and P({X1 <t}) compare when n is large and t is such that t<? c) Compute the odf of Zn = max(X1...., X.). d) How do P({Zn2 t}) and P({X1 2 t}) compare when n is large and t is such...
4. Suppose X1, . . . ,X, are independent, normally distributed with mean E(Xi) and variance...
4. Suppose X1, . . . ,X, are independent, normally distributed with mean E(Xi) and variance Var(X)-σί. Let Żi-(X,-μ.)/oi so that Zi , . . . , Ζ,, are independent and each has a N(0, 1) distribution. Show that LZhas a x2 distribution. Hint: Use the fact that each Z has a xî distribution i naS
(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with...
(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with mean EX1 = i and VarX1 = a2. Let Yı, Y2, ... be another sequence of independent and identically distributed random variables with mean EY = u and VarY1 a2 Define the random variable ( ΣxΣ) 1 Dn 2ng2 i= i=1 Prove that Dn converges in distribution to a standard normal distribution, i.e., prove that 1 P(Dn ) dt 2T as n >oo for...
3. Suppose X1, X2, -- are independent identically distributed random variables with mean 0 and va...
3. Suppose X1, X2, -- are independent identically distributed random variables with mean 0 and variance 1.Let Sn denote the partial sum Let Fn denote the information contained in Xi, .X,. Suppoe m n. (1) Compute El(Sn Sm)lFm (2) Compute ESm(Sn Sm)|F (3) Compute ES|]. (Hint: Write S (4) Verify that S -n is a martingale. [Sm(Sn Sm))2)
3. Suppose X1, X2, -- are independent identically distributed random variables with mean 0 and variance 1.Let Sn denote the partial sum...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
Let Y1<Y2<...<Yn be the order statistics of a random sample of size n from the distribution...
Let Y1<Y2<...<Yn be the
order statistics of a random sample of size n from the distribution
having p.d.f f(x) = e-y , 0<y<, zero elsewhere. Answer the following
questions.
(a) decide whether Z1 = Y2
and Z2=Y4-Y2 are
stochastically independent or not. (hint. first find the joint
p.d.f. of Y2 and Y4)
(b) show that
Z1 = nY1, Z2=
(n-1)(Y2-Y1),
Z3=(n-2)(Y3-Y2), ....,
Zn=Yn-Yn-1
are stocahstically
independent and that each Zi has the exponential
distribution.(hint use change of variable technique)
3. Let X , X, be an independent and identically distributed random sample from a distribution...
3. Let X , X, be an independent and identically distributed random sample from a distribution with pdf f(X; B) = B Xe 4X X20 a. Find the Maximum Likelihood Estimator of B, and denote it ß. b. Find the lower bound for Var() using Cramer-Rao inequality for ſunbiased.
4. Let Xi i = 1,2, ... be independent and identically distributed (i.i.d.) Possion. That is let Px;(k) = Px(k) = 2*24. Let Zn = ??=1 X; and find Pzn. Hint: use characteristic functions. n • Use Li=
Observations X1,..., Xn are independent identically distributed, following the PDF fx:(xi) = 0x8-1, and that 0<Xi <1 for all i. The parameter is an unknown positive number. Find the ML estimator of e
6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when
6.7. Let X,, be a sequence of independent and identically distributed X, and show Pl random variables with mean 0 and variance σ. Let 1-1 that {Z., n 2 1j is a martingale when
8. Let X1...., X, be i.i.d. ~E(1) random variables (i.e., they are independent and identically distributed, all with the exponential distribution of parameter 1 = 1). a) Compute the cdf of Yn = min(X1,...,xn). b) How do P({Y, St}) and P({X1 <t}) compare when n is large and t is such that t<? c) Compute the odf of Zn = max(X1...., X.). d) How do P({Zn2 t}) and P({X1 2 t}) compare when n is large and t is such...
4. Suppose X1, . . . ,X, are independent, normally distributed with mean E(Xi) and variance Var(X)-σί. Let Żi-(X,-μ.)/oi so that Zi , . . . , Ζ,, are independent and each has a N(0, 1) distribution. Show that LZhas a x2 distribution. Hint: Use the fact that each Z has a xî distribution i naS
(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with mean EX1 = i and VarX1 = a2. Let Yı, Y2, ... be another sequence of independent and identically distributed random variables with mean EY = u and VarY1 a2 Define the random variable ( ΣxΣ) 1 Dn 2ng2 i= i=1 Prove that Dn converges in distribution to a standard normal distribution, i.e., prove that 1 P(Dn ) dt 2T as n >oo for...
3. Suppose X1, X2, -- are independent identically distributed random variables with mean 0 and variance 1.Let Sn denote the partial sum Let Fn denote the information contained in Xi, .X,. Suppoe m n. (1) Compute El(Sn Sm)lFm (2) Compute ESm(Sn Sm)|F (3) Compute ES|]. (Hint: Write S (4) Verify that S -n is a martingale. [Sm(Sn Sm))2)
3. Suppose X1, X2, -- are independent identically distributed random variables with mean 0 and variance 1.Let Sn denote the partial sum...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...
Let Y1<Y2<...<Yn be the
order statistics of a random sample of size n from the distribution
having p.d.f f(x) = e-y , 0<y<, zero elsewhere. Answer the following
questions.
(a) decide whether Z1 = Y2
and Z2=Y4-Y2 are
stochastically independent or not. (hint. first find the joint
p.d.f. of Y2 and Y4)
(b) show that
Z1 = nY1, Z2=
(n-1)(Y2-Y1),
Z3=(n-2)(Y3-Y2), ....,
Zn=Yn-Yn-1
are stocahstically
independent and that each Zi has the exponential
distribution.(hint use change of variable technique)
3. Let X , X, be an independent and identically distributed random sample from a distribution with pdf f(X; B) = B Xe 4X X20 a. Find the Maximum Likelihood Estimator of B, and denote it ß. b. Find the lower bound for Var() using Cramer-Rao inequality for ſunbiased.
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