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2. Approximate the following probabilities for sufficiently large n by applying CLT. Please provide answers using...
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1),...
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
(b) For n = 100, give an approximaation for P(Y> 100) (c) Let X be the...
(b) For n = 100, give an approximaation for P(Y> 100) (c) Let X be the sample mean, then approximate P(1.1< 1.2) for -100. 2. Consider a random sample XX from CDF F(a) 1-1/ for z [1, 0o) and zero otherwise. (a) Find the limiting distribution of XiI.n, the smallest order statistic. (b) Find the limiting distribution of XI (c) Find the limiting distribution of n In X1:m- 3. Suppose that X,,, are iid. N(0,o2). Find a function of T(x)x...
iid Let X1,, X, ^ X~P for some unknown distribution P with continuous cdf F. Below...
iid Let X1,, X, ^ X~P for some unknown distribution P with continuous cdf F. Below we describe a ? test for the null and alternative hypotheses We divide the sample space into 5 disjoint subsets refered to as bins A1(-00,-2), A2 -(-2,-0.5), As -(-0.5,0.5), A4 (0.5,2) As -(2, oo). as functions of X, by Now, define discrete random variables For example, if Xi --0.1, then Xi є Аз and so Y;-3. In other words, Y, is the label of...
1. Let 2 ~ N (0,1). Using a standard normal table, find the following probabilities. You...
1. Let 2 ~ N (0,1). Using a standard normal table, find the following probabilities. You do not need to provide any equation. Instead, draw pictures as we did in the lecture and find the numbers from the table. Make yourself be familiar with using different kinds of tables. (Hint: The standard normal density is symmetric around zero.] (a) P(Z < 0) (b) P(Z < 1.96) (c) P(Z < 1.96) (d) P(Z = 1.96) (e) P(-1.65 < 2 <0) (f)...
1. Let X be an iid sample of size n from a continuous distribution with mean...
1. Let X be an iid sample of size n from a continuous distribution with mean /i, variance a2 and such that Xi e [0, 1] for all i e {1,...,n}. Let X = average. For a E (0,1), we wish to obtain a number q > 0 such that: (1/n) Xi be the sample Р(X € |и — 9. и + q) predict with probability approximately In other words, we wish to sample of size n, the average X...
9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,...
9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,x,Je {0,1)、[i.e., X is a binary sequence of length n] and Y-(Y Rt [i.e. Y is a sequence of n real numbers.) ·X1~" Bernoulli(1/2) ,%) E Ip is the switching probability; when p is small the Markov chain likes to stay in the same state] . conditioned on X, the random variables Yı , . . . , y, are...
Let N be a binomial random variable with n = 2 trials and success probability p...
Let N be a binomial random variable with n = 2 trials and success probability p = 0.5. Let X and Y be uniform random variables on [0, 1] and that X, Y, N are mutually independent. Find the probability density function for Z = NXY.
I don't understand a iii and b ii, What's the procedure of deriving the limit distribution? Thanks. 6. Extreme...
I don't understand a iii and b ii, What's the procedure of
deriving the limit distribution? Thanks.
6. Extreme values are of central importance in risk management and the following two questions provide the fundamental tool used in the extreme value theory. (a) Let Xi,... , Xn be independent identically distributed (i. i. d.) exp (1) random variables and define max(Xi,..., Xn) (i) Find the cumulative distribution of Zn (ii) Calculate the cumulative distribution of Vn -Zn - Inn (iii)...
5.2.5 (Example 5.2.6 Continued) Suppose thatXY are iid having the following common distribution. PC,-1-cip.i-1. 2. 3....
5.2.5 (Example 5.2.6 Continued) Suppose thatXY are iid having the following common distribution. PC,-1-cip.i-1. 2. 3. and 2 <p < 3 Here. c c(p) (> 0) is such that Σ | P(X,-i) = 1.. Is there a real number a = a(p) such that Xn → a as n → 00, for all fixed 2 <p < 3? FYI: Example 5.2.6 In order to appreciate the importance of Khinchine's WLLN (Theorem 5.2.3). let us consider a sequence of iid random...
CHOOSE the correct answer and WRITE the corresponding letter in the provided space. 7. If X-Uniform(0.c),...
CHOOSE the correct answer and WRITE the corresponding letter in the provided space. 7. If X-Uniform(0.c), which of the following is CORRECT? A. E[X] = 0. B. P[X S c/2] = 1. C. Fx(c/2) = 0.5. D. fx(c/2) <fx©). 8. Assuming X is a continuous random variable, which of the following is INCORRECT? A x(x) = dFx(x)/dx. B. x(x) dx = 1. C fx(x) dx = Fzco). D. none of the above. 9. The CLT states that for Z =...
Dr. Beldi Qiang STATWOB Flotllework #1 1. Let X.,No X~ be a i.İ.d sample form Exp(1), and Y-Σ-x. (a) Use CLT to get a large sample distribution of Y (b) For n 100, give an approximation for P(Y> 100) (c) Let X be the sample mean, then approximate P(.IX <1.2) for n 100. x, from CDF F(r)-1-1/z for 1 e li,00) and ,ero 2Consider a random sample Xi.x, 、 otherwise. (a) Find the limiting distribution of Xim the smallest order...
(b) For n = 100, give an approximaation for P(Y> 100) (c) Let X be the sample mean, then approximate P(1.1< 1.2) for -100. 2. Consider a random sample XX from CDF F(a) 1-1/ for z [1, 0o) and zero otherwise. (a) Find the limiting distribution of XiI.n, the smallest order statistic. (b) Find the limiting distribution of XI (c) Find the limiting distribution of n In X1:m- 3. Suppose that X,,, are iid. N(0,o2). Find a function of T(x)x...
iid Let X1,, X, ^ X~P for some unknown distribution P with continuous cdf F. Below we describe a ? test for the null and alternative hypotheses We divide the sample space into 5 disjoint subsets refered to as bins A1(-00,-2), A2 -(-2,-0.5), As -(-0.5,0.5), A4 (0.5,2) As -(2, oo). as functions of X, by Now, define discrete random variables For example, if Xi --0.1, then Xi є Аз and so Y;-3. In other words, Y, is the label of...
1. Let 2 ~ N (0,1). Using a standard normal table, find the following probabilities. You do not need to provide any equation. Instead, draw pictures as we did in the lecture and find the numbers from the table. Make yourself be familiar with using different kinds of tables. (Hint: The standard normal density is symmetric around zero.] (a) P(Z < 0) (b) P(Z < 1.96) (c) P(Z < 1.96) (d) P(Z = 1.96) (e) P(-1.65 < 2 <0) (f)...
1. Let X be an iid sample of size n from a continuous distribution with mean /i, variance a2 and such that Xi e [0, 1] for all i e {1,...,n}. Let X = average. For a E (0,1), we wish to obtain a number q > 0 such that: (1/n) Xi be the sample Р(X € |и — 9. и + q) predict with probability approximately In other words, we wish to sample of size n, the average X...
9. Consider the following hidden Markov model (HMM) (This is the same HMM as in the previous HMM problem): ·X=(x, ,x,Je {0,1)、[i.e., X is a binary sequence of length n] and Y-(Y Rt [i.e. Y is a sequence of n real numbers.) ·X1~" Bernoulli(1/2) ,%) E Ip is the switching probability; when p is small the Markov chain likes to stay in the same state] . conditioned on X, the random variables Yı , . . . , y, are...
I don't understand a iii and b ii, What's the procedure of
deriving the limit distribution? Thanks.
6. Extreme values are of central importance in risk management and the following two questions provide the fundamental tool used in the extreme value theory. (a) Let Xi,... , Xn be independent identically distributed (i. i. d.) exp (1) random variables and define max(Xi,..., Xn) (i) Find the cumulative distribution of Zn (ii) Calculate the cumulative distribution of Vn -Zn - Inn (iii)...
5.2.5 (Example 5.2.6 Continued) Suppose thatXY are iid having the following common distribution. PC,-1-cip.i-1. 2. 3. and 2 <p < 3 Here. c c(p) (> 0) is such that Σ | P(X,-i) = 1.. Is there a real number a = a(p) such that Xn → a as n → 00, for all fixed 2 <p < 3? FYI: Example 5.2.6 In order to appreciate the importance of Khinchine's WLLN (Theorem 5.2.3). let us consider a sequence of iid random...
CHOOSE the correct answer and WRITE the corresponding letter in the provided space. 7. If X-Uniform(0.c), which of the following is CORRECT? A. E[X] = 0. B. P[X S c/2] = 1. C. Fx(c/2) = 0.5. D. fx(c/2) <fx©). 8. Assuming X is a continuous random variable, which of the following is INCORRECT? A x(x) = dFx(x)/dx. B. x(x) dx = 1. C fx(x) dx = Fzco). D. none of the above. 9. The CLT states that for Z =...
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