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Problem 6: 10 points A) Find moments of first and second order and...
Problem 5: 10 points Consider n independent variables, {X1, X2,... , Xn) uniformly distributed over the...
Problem 5: 10 points Consider n independent variables, {X1, X2,... , Xn) uniformly distributed over the unit interval, (0,1) Introduce two new random variables, M-max (X1, X2,..., Xn) and N -min (X1, X2,..., Xn) 1. Find the joint distribution of a pair (M,N) 2. Derive the CDF and density for M 3. Derive the CDF and density for N.
(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)...
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...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z max (X. Y) as the larger of the two, Derive the C.DF. and density function for Z. 2. Define W min(X,Y) as the smaller of the two. Derive the C.D.F.and density function for W 3. Derive the joint density of the pair (W. Z). Specify where the density if positive and where it takes a zero value....
Consider n independent and identically distributed random variables X1,X2, following a uniform distribution on the interval...
Consider n independent and identically distributed random variables X1,X2, following a uniform distribution on the interval [0,1] ,Xn, each a) What is the pdf of Mmin(X1,X2, .. ,Xn)? b) Give the expectation and variance of XX 1-1лі.
Need only parts 5 and 6 Problem 6: 10 points Assume that X and Y are...
Need only parts 5 and 6
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0, 1) 1. Define Z = max (X, Y) as the larger of the two. Derive the CD. F. and density function for Z 2. Define W- min (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W. 3. Derive the joint density of the pair (W, Z). Specify...
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)"
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z-max (X, Y) as the larger of the two. Derive the C.D.F. and density function for Z. 2. Define Wmin (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W 3. Derive the joint density of the pair (W, Z). Specify where the density if positive and where it takes a zero...
l. X) points Lei Xi, X, X b e random variables . I. adl X, is...
l. X) points Lei Xi, X, X b e random variables . I. adl X, is "uifornly disi rilnicd 。" on [0,1]. The random variables Xi, X2, X3,... are independent. The random variable N is the first integer n 2 1 such that Xn 2 c where 0< c< is a constant. That is, N = min(n : Xn-c). What is EM?
3.9. Problem*. (Section 9.1) The following problems concern maximums and minimums of collections of independent random...
3.9. Problem*. (Section 9.1) The following problems concern maximums and minimums of collections of independent random variables. (a) Let Y.Y2, ..., Yn be independent exponential random variables with parameters 11, 12,..., In, respectively. Prove that E[min{Yı, Y2, ..., Yn}] < min{E[Y], E[Y2),..., E|Y.]} (b) Suppose that X1, X2, ..., X, are independent continuous random variables with uni- form distributions on (0,1). Compute E[min{X1, X2, ..., Xn}] and E[max{X1, X2,..., X.}]
Problem 5: 10 points Consider n independent variables, {X1, X2,... , Xn) uniformly distributed over the unit interval, (0,1) Introduce two new random variables, M-max (X1, X2,..., Xn) and N -min (X1, X2,..., Xn) 1. Find the joint distribution of a pair (M,N) 2. Derive the CDF and density for M 3. Derive the CDF and density for N.
(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)...
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...
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z max (X. Y) as the larger of the two, Derive the C.DF. and density function for Z. 2. Define W min(X,Y) as the smaller of the two. Derive the C.D.F.and density function for W 3. Derive the joint density of the pair (W. Z). Specify where the density if positive and where it takes a zero value....
Consider n independent and identically distributed random variables X1,X2, following a uniform distribution on the interval [0,1] ,Xn, each a) What is the pdf of Mmin(X1,X2, .. ,Xn)? b) Give the expectation and variance of XX 1-1лі.
Need only parts 5 and 6
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0, 1) 1. Define Z = max (X, Y) as the larger of the two. Derive the CD. F. and density function for Z 2. Define W- min (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W. 3. Derive the joint density of the pair (W, Z). Specify...
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)"
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z-max (X, Y) as the larger of the two. Derive the C.D.F. and density function for Z. 2. Define Wmin (X, Y) as the smaller of the two. Derive the C.D.F. and density function for W 3. Derive the joint density of the pair (W, Z). Specify where the density if positive and where it takes a zero...
l. X) points Lei Xi, X, X b e random variables . I. adl X, is "uifornly disi rilnicd 。" on [0,1]. The random variables Xi, X2, X3,... are independent. The random variable N is the first integer n 2 1 such that Xn 2 c where 0< c< is a constant. That is, N = min(n : Xn-c). What is EM?
3.9. Problem*. (Section 9.1) The following problems concern maximums and minimums of collections of independent random variables. (a) Let Y.Y2, ..., Yn be independent exponential random variables with parameters 11, 12,..., In, respectively. Prove that E[min{Yı, Y2, ..., Yn}] < min{E[Y], E[Y2),..., E|Y.]} (b) Suppose that X1, X2, ..., X, are independent continuous random variables with uni- form distributions on (0,1). Compute E[min{X1, X2, ..., Xn}] and E[max{X1, X2,..., X.}]
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