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4. Let X,X,Bernoulli(p), and let Y Xi. Then we know that Y-Binomial(n.p) Consider the hypotheses Hop-Po...
LetX,X2, , XnLLd. Bernoulli(p), and let Y-Σ,Xi. Then we know that Y-Binomial(n, p). 5. Consider t...
LetX,X2, , XnLLd. Bernoulli(p), and let Y-Σ,Xi. Then we know that Y-Binomial(n, p). 5. Consider the hypotheses Hop-po against HA:p#po- a. Find the likelihood function of p in terms of random variable Y, L(p). b. Construct the (generalized) likelihood ratio λ(v). Hint: what is pMLE?] C. (i) For the particular case of po 0.25 and n 5, fill in the table: 3 4 A(y) (ii) Rearrange the table in the order of increasing of values of 2, and compute cumulative...
Problem 6: Suppose we observe a random variable X having a binomial distribution with parameters n and zp. (a) What is...
Problem 6: Suppose we observe a random variable X having a binomial distribution with parameters n and zp. (a) What is the generalized likelihood ratio for testing Ho : p-0.5 against H, : p* 0.5? (b) Show that a generalized likelihood ratio test rejects Ho when |X -n/2|2 c. (Hint: it may help to consider the logarithm of the generalized likelihood ratio.) (c) What is the significance level of the test when n 12 and c 5?
Problem 6: Suppose...
1(a) Let Xi, X2, the random interval (ay,, b%) around 9, where Y, = max(Xi,X2 ,X),...
1(a) Let Xi, X2, the random interval (ay,, b%) around 9, where Y, = max(Xi,X2 ,X), a and b are constants such that 1 S a <b. Find the confidence level of this interval. Xi, X, want to test H0: θ-ya versus H1: θ> %. Suppose we set our decision rule as reject Ho , X, be a random sample from the Uniform (0, θ) distribution. Consider (b) ,X5 is a random sample from the Bernoulli (0) distribution, 0 <...
Suppose that we have two independent binomial random variables X ~Binomial(n, px) and Y ~ Binomial(m,Pv)....
Suppose that we have two independent binomial random variables X ~Binomial(n, px) and Y ~ Binomial(m,Pv). You can assume that the MLE's are -X/n and p,-Y/m. (a) Find the MLE for p under the assumption that p (b) Find the LRT statistic T for testing p,-py HA:p.Ру vs. (c) Evaluate the value of this statistic if n 353, X 95, m -432, and Y 123. (d) Compare the answer from part (c) to a critical value from a x2 with...
Exercises 10.3. Let Xi . . . , x N μ, σ2), whereơ2 s known to be equal to 100. In testing Ho : 25...
Exercises 10.3. Let Xi . . . , x N μ, σ2), whereơ2 s known to be equal to 100. In testing Ho : 25vs. H :H>25,h What sample size n would be necessary if one wishes to reject Ho with probability at least 95 if μ 26? iid se that a coin is to be tossed n times, and you wish to test the hypothesis Ho:p-12 VS. Hi P> I/2 at a- .05. What sample size n would be...
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...
3. Suppose that we have two independent binomial random variablesXBinomial(n,pr) and Y~Binomial(m, p). You can assume...
3. Suppose that we have two independent binomial random variablesXBinomial(n,pr) and Y~Binomial(m, p). You can assume that the MLE's are p X/n and py- Y/m. (a) Find the MLE for p under the assumption that pr -Py (b) Find the LRT statistic T for testing (c) Evaluate the value of this statistic if n 353, X 95, m 432, and Y 123. (d) Compare the answer from part (c) to a critical value from a x2 with one degree of...
Let X, Y be independent random variables where X is binomial(n = 4, p = 1/3)...
Let X, Y be independent random variables where X is binomial(n = 4, p = 1/3) and Y is binomial(n = 3,p = 1/3). Find the moment-generating functions of the three random variables X, Y and X + Y . (You may look up the first two. The third follows from the first two and the behavior of moment-generating functions.) Now use the moment-generating function of X + Y to find the distribution of X + Y .
(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...
B(1.p.) and Y.Y,....Yo be a r.s. from 4. Let X1, X2-... X be a r.s. from...
B(1.p.) and Y.Y,....Yo be a r.s. from 4. Let X1, X2-... X be a r.s. from B(1,Py). We wish to test HOP: = Py vs. H:P: Py- (a) Find test statistic and its distribution under Ho. (b) Find the rejection region at the significant level of a = 0.05. (c) What is your decision when you have X = 34, _X Y = 19, Y = 19, n = 40, and m = 35 from the data? = 34,
LetX,X2, , XnLLd. Bernoulli(p), and let Y-Σ,Xi. Then we know that Y-Binomial(n, p). 5. Consider the hypotheses Hop-po against HA:p#po- a. Find the likelihood function of p in terms of random variable Y, L(p). b. Construct the (generalized) likelihood ratio λ(v). Hint: what is pMLE?] C. (i) For the particular case of po 0.25 and n 5, fill in the table: 3 4 A(y) (ii) Rearrange the table in the order of increasing of values of 2, and compute cumulative...
Problem 6: Suppose we observe a random variable X having a binomial distribution with parameters n and zp. (a) What is the generalized likelihood ratio for testing Ho : p-0.5 against H, : p* 0.5? (b) Show that a generalized likelihood ratio test rejects Ho when |X -n/2|2 c. (Hint: it may help to consider the logarithm of the generalized likelihood ratio.) (c) What is the significance level of the test when n 12 and c 5?
Problem 6: Suppose...
1(a) Let Xi, X2, the random interval (ay,, b%) around 9, where Y, = max(Xi,X2 ,X), a and b are constants such that 1 S a <b. Find the confidence level of this interval. Xi, X, want to test H0: θ-ya versus H1: θ> %. Suppose we set our decision rule as reject Ho , X, be a random sample from the Uniform (0, θ) distribution. Consider (b) ,X5 is a random sample from the Bernoulli (0) distribution, 0 <...
Suppose that we have two independent binomial random variables X ~Binomial(n, px) and Y ~ Binomial(m,Pv). You can assume that the MLE's are -X/n and p,-Y/m. (a) Find the MLE for p under the assumption that p (b) Find the LRT statistic T for testing p,-py HA:p.Ру vs. (c) Evaluate the value of this statistic if n 353, X 95, m -432, and Y 123. (d) Compare the answer from part (c) to a critical value from a x2 with...
Exercises 10.3. Let Xi . . . , x N μ, σ2), whereơ2 s known to be equal to 100. In testing Ho : 25vs. H :H>25,h What sample size n would be necessary if one wishes to reject Ho with probability at least 95 if μ 26? iid se that a coin is to be tossed n times, and you wish to test the hypothesis Ho:p-12 VS. Hi P> I/2 at a- .05. What sample size n would be...
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...
3. Suppose that we have two independent binomial random variablesXBinomial(n,pr) and Y~Binomial(m, p). You can assume that the MLE's are p X/n and py- Y/m. (a) Find the MLE for p under the assumption that pr -Py (b) Find the LRT statistic T for testing (c) Evaluate the value of this statistic if n 353, X 95, m 432, and Y 123. (d) Compare the answer from part (c) to a critical value from a x2 with one degree of...
(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...
B(1.p.) and Y.Y,....Yo be a r.s. from 4. Let X1, X2-... X be a r.s. from B(1,Py). We wish to test HOP: = Py vs. H:P: Py- (a) Find test statistic and its distribution under Ho. (b) Find the rejection region at the significant level of a = 0.05. (c) What is your decision when you have X = 34, _X Y = 19, Y = 19, n = 40, and m = 35 from the data? = 34,
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