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Problem 1: The goal of this problem is to show that if X has a N(0,...
3. Consider a discrete random variable X which follows the geometric distribution f(x,p) = pr-1(1...
3. Consider a discrete random variable X which follows the geometric distribution f(x,p) = pr-1(1-p), x = 1.2. . . . , 0 < p < 1. Recall that E(x) (1-p) (a) Find the Fisher information I(p). (b) Show that the Cramer-Rao inequality is strict e) Let XX ~X. Find the maximum likelihood estimator of p. Note that the expression you find may look complicated and hard to evaluate. (d) Now modify your view by setting μ T1p such that...
(Sheldon Ross) Consider a process {X,, п : 0, 1, . ( 1, 2, 31, suppose ..1, which takes on the values aij n even, Pj nodd, where j-1 for i = 1, 2, 3. Is {X, | n > 0} a Markov chain ? If not, show...
(Sheldon Ross) Consider a process {X,, п : 0, 1, . ( 1, 2, 31, suppose ..1, which takes on the values aij n even, Pj nodd, where j-1 for i = 1, 2, 3. Is {X, | n > 0} a Markov chain ? If not, show how by enlarging the state space, we may transform it into a Markov chain
(Sheldon Ross) Consider a process {X,, п : 0, 1, . ( 1, 2, 31, suppose ..1, which...
(5) Show that if t,... ., are positive numbers such that ty+..+tn-1 and .. n E (0,1/2], then n. F...
(5) Show that if t,... ., are positive numbers such that ty+..+tn-1 and .. n E (0,1/2], then n. First check that the function f(x) In (1-)-n(x) is convex on the interval (0, 1/2). The inequality is due to Ky Fan.]
(5) Show that if t,... ., are positive numbers such that ty+..+tn-1 and .. n E (0,1/2], then n. First check that the function f(x) In (1-)-n(x) is convex on the interval (0, 1/2). The inequality is due to...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y)...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
Example 1: X is uniformly distributed over |-1.1]. Calculate P(X1 2a) where a 0 using the...
Example 1: X is uniformly distributed over |-1.1]. Calculate P(X1 2a) where a 0 using the following 3 methods (a) The exact method (b) Markov inequality (c) Chebyshev inequality
2. Consider the following minimization problem minf(x) e - COS T on 0, 1]. Find the...
2. Consider the following minimization problem minf(x) e - COS T on 0, 1]. Find the minimizer using Golden section method with e 1/2 by hand. (10pt)
2. Consider the following minimization problem minf(x) e - COS T on 0, 1]. Find the minimizer using Golden section method with e 1/2 by hand. (10pt)
2. Suppose that X Binom(n,p) such that n>1 and 0 <p<1. Show that E[(x + 1)-1...
2. Suppose that X Binom(n,p) such that n>1 and 0 <p<1. Show that E[(x + 1)-1 = _(1 – p)p+1 – 1 p(n + 1)
Many thanks!! (a) Let fn(x) max(1 - |x -n|,0) for each n 2 1. Show that...
Many thanks!!
(a) Let fn(x) max(1 - |x -n|,0) for each n 2 1. Show that {fn} is a bounded sequence in LP (R) for all p E [1, 00]. Show that fn >0 pointwise everywhere in R, i.e. fn(x) -> 0 for all x E R. Show that fn does not converge to 0 in LP (R) (b) Fix p E 1, o0). Let fn E LP(0, 1) be defined by fn(x) n1/? on [0,1/n), and fn(x)0 otherwise. Show...
Help please! Let {Xn}n=0 be a process taking values in a countable [0, 1]E and stochastic...
Help please!
Let {Xn}n=0 be a process taking values in a countable [0, 1]E and stochastic set E, and assume that for some probability vector X matriz P E(0, 1ExE we have prove that Xn ~ Markov(λ, P)
Let Xo, X1, n 0, 1, 2, . . . . Show that YO, Yı ,...
Let Xo, X1, n 0, 1, 2, . . . . Show that YO, Yı , matrix ,... be a Markov chain with transition matrix P. Let Yn - X3n, for is a Markov chain and exhibit its transition
3. Consider a discrete random variable X which follows the geometric distribution f(x,p) = pr-1(1-p), x = 1.2. . . . , 0 < p < 1. Recall that E(x) (1-p) (a) Find the Fisher information I(p). (b) Show that the Cramer-Rao inequality is strict e) Let XX ~X. Find the maximum likelihood estimator of p. Note that the expression you find may look complicated and hard to evaluate. (d) Now modify your view by setting μ T1p such that...
(Sheldon Ross) Consider a process {X,, п : 0, 1, . ( 1, 2, 31, suppose ..1, which takes on the values aij n even, Pj nodd, where j-1 for i = 1, 2, 3. Is {X, | n > 0} a Markov chain ? If not, show how by enlarging the state space, we may transform it into a Markov chain
(Sheldon Ross) Consider a process {X,, п : 0, 1, . ( 1, 2, 31, suppose ..1, which...
(5) Show that if t,... ., are positive numbers such that ty+..+tn-1 and .. n E (0,1/2], then n. First check that the function f(x) In (1-)-n(x) is convex on the interval (0, 1/2). The inequality is due to Ky Fan.]
(5) Show that if t,... ., are positive numbers such that ty+..+tn-1 and .. n E (0,1/2], then n. First check that the function f(x) In (1-)-n(x) is convex on the interval (0, 1/2). The inequality is due to...
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
Example 1: X is uniformly distributed over |-1.1]. Calculate P(X1 2a) where a 0 using the following 3 methods (a) The exact method (b) Markov inequality (c) Chebyshev inequality
2. Consider the following minimization problem minf(x) e - COS T on 0, 1]. Find the minimizer using Golden section method with e 1/2 by hand. (10pt)
2. Consider the following minimization problem minf(x) e - COS T on 0, 1]. Find the minimizer using Golden section method with e 1/2 by hand. (10pt)
2. Suppose that X Binom(n,p) such that n>1 and 0 <p<1. Show that E[(x + 1)-1 = _(1 – p)p+1 – 1 p(n + 1)
Many thanks!!
(a) Let fn(x) max(1 - |x -n|,0) for each n 2 1. Show that {fn} is a bounded sequence in LP (R) for all p E [1, 00]. Show that fn >0 pointwise everywhere in R, i.e. fn(x) -> 0 for all x E R. Show that fn does not converge to 0 in LP (R) (b) Fix p E 1, o0). Let fn E LP(0, 1) be defined by fn(x) n1/? on [0,1/n), and fn(x)0 otherwise. Show...
Help please!
Let {Xn}n=0 be a process taking values in a countable [0, 1]E and stochastic set E, and assume that for some probability vector X matriz P E(0, 1ExE we have prove that Xn ~ Markov(λ, P)
Let Xo, X1, n 0, 1, 2, . . . . Show that YO, Yı , matrix ,... be a Markov chain with transition matrix P. Let Yn - X3n, for is a Markov chain and exhibit its transition
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