Please answer all the parts neatly with all details.
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Please answer all the parts neatly with all details. 3. Assume X1, X2,... are a sequence...
Please answer all the parts neatly with all details. 4. Let X1, X2,... be independent random...
Please answer all the parts neatly with all details.
4. Let X1, X2,... be independent random variables satisfying E(X4) < B for some finite B > 0 and E(Xn)-> . (a) Show that Y = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) < B, E(Y4 (b) Show that for Y, = (Yi +..+ Y)/n, 16B 16B ΣειβΥ< 6B 1 = n4 i=1 6 + n4 ij ΣΕ Υ) E(Y4) n'3 n2 P(Y > €) < oo and...
Please answer all the parts neatly with all details. 8. Let X1, X2, ... be an...
Please answer all the parts neatly with all details.
8. Let X1, X2, ... be an i.i.d. with Xn Let Y min(X,... , Xn) + 1 and Zn = max(X1,... , Xn) - 1. (a) Show that Y, - 0 and 0 - Z, have the same distribution uniform(0 1,0+ 1) (b) Show that Y, -P>0. (c) Show that n(Yn - Zn) converges in distribution and specify the limit distribution
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence...
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
Please answer all the parts neatly with all details. 6. Two independent random variables X and...
Please answer all the parts neatly with all details.
6. Two independent random variables X and Y have the same distribution with finite second moment Assume X and (X +Y)/2 have the same distribution (a) Show that the expectation of X is zero. (b) Show that X and (X1 ...+X2n)/V2n have the same distribution where X\, X2,... are independent and have identical distribution of X (c) Show that X~ N(0, 02) for some o2 > 0.
6. Two independent random...
7. Let X1, X2, ... be an i.i.d. random variables. (a) Show that max(X1,... , X,n)/n...
7. Let X1, X2, ... be an i.i.d. random variables. (a) Show that max(X1,... , X,n)/n >0 in probability if nP(Xn > n) -» 0. (b) Find a random variable Y satisfying nP(Y > n) ->0 and E(Y) = Oo
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e...
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e of tnduqendent idente onm the interval (o, 1] and let Compute the (almost surely) limit of Yn (b) (5 points) Let X1, X2,... be independent randon variables such that Xn is a discrete random variable uniform on the set {1, 2, . . . , n + 1]. Let Yn = min(X1,X2, . . . , Xn} be the smallest value among Xj,Xn. Show...
(7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi , . . . , X,.), V=min(X1, ,X,). (a) Find the distribution functio...
(7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi , . . . , X,.), V=min(X1, ,X,). (a) Find the distribution function and the density function of U and of V (b) Show that the joint density function of U and V is fe,y(u, u)= n(n-1)/(u)/(v)[F(v)-F(u)]n-1, ifu < u.
(7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi...
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the...
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b) Find E(V) and E(W)
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b)...
4. Let X1, X2, . .. be independent random variables satisfying E(X) E(Xn) --fi. (a) Show...
4. Let X1, X2, . .. be independent random variables satisfying E(X) E(Xn) --fi. (a) Show that Y, = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) (b) Show that for Y, = (Y1 + . . + Y,)/n, <B for some finite B > 0 and VB,E(Y) < 16B. 16B 6B 1 E(Y) E(Y) n4 i1 n4 n3 (c) Show that P(Y, > e) < 0 and conclude Y, ->0 almost surely (d) Show that (i1 +...
5. (4 points) Let X1, X2, be independent random variables that are uniformly distributed on [-1,1]...
5. (4 points) Let X1, X2, be independent random variables that are uniformly distributed on [-1,1] Show that the sequence Yi,Y2, converges in probability to some limit, and identity the limit, for each of the following cases: (a) Yn = max Xi, , xn (this is similar to an example from class). (c) Yn = (Xn)"
Please answer all the parts neatly with all details.
4. Let X1, X2,... be independent random variables satisfying E(X4) < B for some finite B > 0 and E(Xn)-> . (a) Show that Y = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) < B, E(Y4 (b) Show that for Y, = (Yi +..+ Y)/n, 16B 16B ΣειβΥ< 6B 1 = n4 i=1 6 + n4 ij ΣΕ Υ) E(Y4) n'3 n2 P(Y > €) < oo and...
Please answer all the parts neatly with all details.
8. Let X1, X2, ... be an i.i.d. with Xn Let Y min(X,... , Xn) + 1 and Zn = max(X1,... , Xn) - 1. (a) Show that Y, - 0 and 0 - Z, have the same distribution uniform(0 1,0+ 1) (b) Show that Y, -P>0. (c) Show that n(Yn - Zn) converges in distribution and specify the limit distribution
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
Please answer all the parts neatly with all details.
6. Two independent random variables X and Y have the same distribution with finite second moment Assume X and (X +Y)/2 have the same distribution (a) Show that the expectation of X is zero. (b) Show that X and (X1 ...+X2n)/V2n have the same distribution where X\, X2,... are independent and have identical distribution of X (c) Show that X~ N(0, 02) for some o2 > 0.
6. Two independent random...
7. Let X1, X2, ... be an i.i.d. random variables. (a) Show that max(X1,... , X,n)/n >0 in probability if nP(Xn > n) -» 0. (b) Find a random variable Y satisfying nP(Y > n) ->0 and E(Y) = Oo
3. (a) (5 points) Let Xi,... be a sequence of independent identically distributed random variables e of tnduqendent idente onm the interval (o, 1] and let Compute the (almost surely) limit of Yn (b) (5 points) Let X1, X2,... be independent randon variables such that Xn is a discrete random variable uniform on the set {1, 2, . . . , n + 1]. Let Yn = min(X1,X2, . . . , Xn} be the smallest value among Xj,Xn. Show...
(7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi , . . . , X,.), V=min(X1, ,X,). (a) Find the distribution function and the density function of U and of V (b) Show that the joint density function of U and V is fe,y(u, u)= n(n-1)/(u)/(v)[F(v)-F(u)]n-1, ifu < u.
(7) Let X1,Xn are i.i.d. random variables, each with probability distribution F and prob- ability density function f. Define U=max{Xi...
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b) Find E(V) and E(W)
(5) Let X1,X2,,Xn be independent identically distributed (i.i.d.) random variables from 1.1 U(0,1). Denote V max{Xi,..., Xn) and W min{Xi,..., Xn] (a) Find the distributions and the densities and the distributions of each of V and W. (b)...
4. Let X1, X2, . .. be independent random variables satisfying E(X) E(Xn) --fi. (a) Show that Y, = Xn - E(Xn) are independent and E(Yn) = 0, E(Y2) (b) Show that for Y, = (Y1 + . . + Y,)/n, <B for some finite B > 0 and VB,E(Y) < 16B. 16B 6B 1 E(Y) E(Y) n4 i1 n4 n3 (c) Show that P(Y, > e) < 0 and conclude Y, ->0 almost surely (d) Show that (i1 +...
5. (4 points) Let X1, X2, be independent random variables that are uniformly distributed on [-1,1] Show that the sequence Yi,Y2, converges in probability to some limit, and identity the limit, for each of the following cases: (a) Yn = max Xi, , xn (this is similar to an example from class). (c) Yn = (Xn)"
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