Homework Answers
Here is define what martingale is. I use the fact that Y's are iid an do the problem.
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2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are...
In the following questions, let Bt denote a Brownian motion with B0 = 0.
In the following questions, let Bt denote a Brownian motion with
B0 = 0.
Let Xn, n-0, 1, 2, denote a biased random walk given by X0 0 and Xn+1 = Xn+Yn+1, where {Yn} are i.i.d. random variables with N(-1,1) distribution. Show that MnX +2nXn +n(n - 1) is a martingale.
Let Xn, n-0, 1, 2, denote a biased random walk given by X0 0 and Xn+1 = Xn+Yn+1, where {Yn} are i.i.d. random variables with N(-1,1) distribution. Show that...
Let Xn is a simple random walk (p = 1/2) on {0, 1, · · · , 100} with absorbing bound- aries. Supp...
Let Xn is a simple random walk (p = 1/2) on {0, 1, · · · , 100} with absorbing bound- aries. Suppose X0 = 50. Let T = min{j : Xj = 0 or 100}. Let Fn denote the information contained in X1,··· ,Xn. (1) Verify that Xn is a martingale. (2) Find P (XT = 100). (3) Let Mn = Xn2 − n. Verify that Mn is also a martingale. (4) It is known that Mn and T...
Let us consider the biased random walk S,-X1 +X2+···+X,, with S: is a sequence of independent...
Let us consider the biased random walk S,-X1 +X2+···+X,, with S: is a sequence of independent randon variables with P(X,--1)-g,P(X.-1) p+q=1 and Mt Show that M.-Sn-b- calculate Elr), where τ = inf{n : S" = a or-6) with a, b > 0. 0. where Xi, X2 p. where g)n is a martingale. Use this martingale to
- Let {Xn} denote a sequence of iid random variables such that P(Xi = 1) =...
- Let {Xn} denote a sequence of iid random variables such that P(Xi = 1) = P(X1 = -1) = 1/2. Let Sn = X1 + X2 + ... + xn. (a) Find ES, and var(Sn); (b) Show that Sn is a martingale.
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information co...
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information contained in X1, ,Xn. (1) Verify that Sn nu is a martingale. (2) Assume that μ 0, verify that Sn-nơ2 is a martingale.
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n...
Let X1, X2, ...,Xn denote a random sample of size n from a Pareto distribution. X(1)...
Let X1, X2, ...,Xn denote a random sample of size n from a Pareto distribution. X(1) = min(X1, X2, ..., Xn) has the cumulative distribution function given by: αη 1 - ( r> B X F(x) = . x <B 0 Show that X(1) is a consistent estimator of ß.
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x...
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)?
8. Let X1,...,Xn denote a random...
Question 5 /10 points/ MLE of Variance is Biased For each n-1.M, let Xn ~ N(μ, Σ) denote an insta...
Question 5 /10 points/ MLE of Variance is Biased For each n-1.M, let Xn ~ N(μ, Σ) denote an instance drawn (independently) from a Gaussian distribution with mean μ and convariance Σ. Recall /IML Xm, and Show that EML]-NN Σ Y ou mav want to prove, then use . where àn,m = 1 if m n and = 0 otherwise.
Question 5 /10 points/ MLE of Variance is Biased For each n-1.M, let Xn ~ N(μ, Σ) denote an instance...
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . ....
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . . TL: n+1 , The stochastic process Xn,n 0, 1,2,3 is a Markov chain, but with a continuous state space. (a) Find EXn and Var(X). (b) Give probability distribution of Xn (c) Find limn oo P(X, > є) for any e> 0. (d) Simulate two realisations of the Markov process from n = 0 until...
(Stochastic process and probability theory) Let Xn, n > 1, denote a sequence of independent random...
(Stochastic process and probability theory) Let Xn, n > 1, denote a sequence of independent random variables with E(Xn) = p. Consider the sequence of random variables În = n(n-1) {x,x, which is an unbiased estimator of up. Does (a) in f H² ? (6) ûn 4* H?? (c) în + k in mean square? (d) Does the estimator în follow a normal distribution if n + ?
In the following questions, let Bt denote a Brownian motion with
B0 = 0.
Let Xn, n-0, 1, 2, denote a biased random walk given by X0 0 and Xn+1 = Xn+Yn+1, where {Yn} are i.i.d. random variables with N(-1,1) distribution. Show that MnX +2nXn +n(n - 1) is a martingale.
Let Xn, n-0, 1, 2, denote a biased random walk given by X0 0 and Xn+1 = Xn+Yn+1, where {Yn} are i.i.d. random variables with N(-1,1) distribution. Show that...
Let us consider the biased random walk S,-X1 +X2+···+X,, with S: is a sequence of independent randon variables with P(X,--1)-g,P(X.-1) p+q=1 and Mt Show that M.-Sn-b- calculate Elr), where τ = inf{n : S" = a or-6) with a, b > 0. 0. where Xi, X2 p. where g)n is a martingale. Use this martingale to
- Let {Xn} denote a sequence of iid random variables such that P(Xi = 1) = P(X1 = -1) = 1/2. Let Sn = X1 + X2 + ... + xn. (a) Find ES, and var(Sn); (b) Show that Sn is a martingale.
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information contained in X1, ,Xn. (1) Verify that Sn nu is a martingale. (2) Assume that μ 0, verify that Sn-nơ2 is a martingale.
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n...
Let X1, X2, ...,Xn denote a random sample of size n from a Pareto distribution. X(1) = min(X1, X2, ..., Xn) has the cumulative distribution function given by: αη 1 - ( r> B X F(x) = . x <B 0 Show that X(1) is a consistent estimator of ß.
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)?
8. Let X1,...,Xn denote a random...
Question 5 /10 points/ MLE of Variance is Biased For each n-1.M, let Xn ~ N(μ, Σ) denote an instance drawn (independently) from a Gaussian distribution with mean μ and convariance Σ. Recall /IML Xm, and Show that EML]-NN Σ Y ou mav want to prove, then use . where àn,m = 1 if m n and = 0 otherwise.
Question 5 /10 points/ MLE of Variance is Biased For each n-1.M, let Xn ~ N(μ, Σ) denote an instance...
4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . . TL: n+1 , The stochastic process Xn,n 0, 1,2,3 is a Markov chain, but with a continuous state space. (a) Find EXn and Var(X). (b) Give probability distribution of Xn (c) Find limn oo P(X, > є) for any e> 0. (d) Simulate two realisations of the Markov process from n = 0 until...
(Stochastic process and probability theory) Let Xn, n > 1, denote a sequence of independent random variables with E(Xn) = p. Consider the sequence of random variables În = n(n-1) {x,x, which is an unbiased estimator of up. Does (a) in f H² ? (6) ûn 4* H?? (c) în + k in mean square? (d) Does the estimator în follow a normal distribution if n + ?
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