Homework Answers
c)yes we can make better
rejection region if we take rejection region as x≥e^1. But in this
case type 1 error increases to some extent.
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2. (17 points total) Let X be a random variable with the following distribution: f(x) 2a...
A random variable, X, has uniform distribution on the interval [0,θ] where θ is unknown. A...
A random variable, X, has uniform distribution on the interval [0,θ] where θ is unknown. A hypothesis test is as follows: H0: θ = 2 H1: θ ≠ 2 It has been decided to reject H0 if the observed value of x is x ≤ 0.1 or x ≥ 1.9. Part a: Find the probability of committing a Type I error. Part b: Suppose the true value of θ is 3. Find the probability of committing a Type II error....
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+ 1 for some integer m. Let Y be the sample median and Z = max(Xi) be the sample maximum (a) Apply the usual formula for the density of an order statistic to show the density of Y is (b) Note that a beta random variable X has density f(x) = TaT(可 with mean μ = α/(a + β) and variance σ2 = αβ/((a +s+...
4. An exponential random variable X has p.d.f (x,07/», x>0, c.df. F(x:0) 1-exp(-x/8) for x >...
4. An exponential random variable X has p.d.f (x,07/», x>0, c.df. F(x:0) 1-exp(-x/8) for x > 0, and mean θ. A single observation of an exponential random variable X is used to test H0 : θ-2 against H1 : θ-5. The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. (a) What is the probability of committing a Type I error? (b) What is the probability of committing a Type...
Consider two pdfs f(C) and f, (x) which are defined as follows 1 0sxs1 f(x)=16 otherwise and i(x)...
Consider two pdfs f(C) and f, (x) which are defined as follows 1 0sxs1 f(x)=16 otherwise and i(x)- 0 otherwise Suppose that a single observation X is to be taken from f(x) where f(x) is either :G(x) or f(x). The following simple hypotheses are to be tested: Ho:(x)- (x) 4: f (x) = f(x) Determine a test procedure δ for which the value of α (6) + 23(6) is minimum. Report both the rejection region and the corresponding minimum value...
Let X1, X2,.,X10 be a sample of size 10 from an exponential distribution with the density function Sae -Xx f(x; A) othe...
Let X1, X2,.,X10 be a sample of size 10 from an exponential distribution with the density function Sae -Xx f(x; A) otherwise 10 We reject Ho : ^ = 1 in favor of H : 1 = 2 if the observed value of Y = smaller than 6 (a) Find the probability of type 1 error for this test. (b) Find the probability of type 2 error for this test (c) Let y5 be the observed value of Y. Find...
Problem H5 Let X be a single observation (n-1) from the following distribution: f(rle)-o elsewhere NOTE:XBeta(0,...
Problem H5 Let X be a single observation (n-1) from the following distribution: f(rle)-o elsewhere NOTE:XBeta(0, 1) The following two hypotheses are being tested: 110 : e-2 vs Ha : ?-1. (a) Draw a graph of f(z | ?) when (i) Ho is true and when (ii) H. is true. Put both graphs on the same plot. Explain why a rejection region of the form (X<k) makes intuitive sense (b) Find k, so that the test has level a 0.05....
3. Let X, X,. X, be a random sample from a distribution with density f(x)- )...
3. Let X, X,. X, be a random sample from a distribution with density f(x)- ) if 0 < x < θ 0 otherwise (a) (b) Determine the density of =max(X1,X2, ,Xn} Use the result of (a) to show that θ is a biased estimator of θ. Determine the bias, B(6) Calculate the mean-square error, MSE(6). (c)
3. Let X1, X2, . . . , Xn be independent samples of a random variable...
3. Let X1, X2, . . . , Xn be independent samples of a random variable with the probability density function (PDF): fX(x) = θ(x − 1/ 2 ) + 1, 0 ≤ x ≤ 1 ,0 otherwise where θ ∈ [−2, 2] is an unknown parameter. We define the estimator ˆθn = 12X − 6 to estimate θ. (a) Is ˆθn an unbiased estimator of θ? (b) Is ˆθn a consistent estimator of θ? (c) Find the mean squared...
MA2500/18 8. Let X be a random variable and let 'f(r; θ) be its PDF where...
MA2500/18 8. Let X be a random variable and let 'f(r; θ) be its PDF where θ is an unknown scalar parameter. We wish to test the simple null hypothesis Ho: 0 against the simple alternative Hi : θ-64. (a) Define the simple likelihood ratio test (SLRT) of Ho against H (b) Show that the SLRT is a most powerful test of Ho against H. (c) Let Xi, X2.... , X be a random sample of observations from the Poisson(e)...
Problem 4 Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0 otherwise Let X1, … Xn be iid random variables with density f for some unknown θ...
Problem 4 Define f(x) as follows θ2 -1<=x<0 1-θ2 0<=x>1 0 otherwise Let X1, … Xn be iid random variables with density f for some unknown θ (0,1), Let a be the number of Xi which are negatives and b be the number of Xi which are positive. Total number of samples n = a+b. Find he Maximum likelihood estimator of θ? Is it asymptotically normal in this sample? Find the asymptotic variance Consider the following hypotheses: H0: X is...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+ 1 for some integer m. Let Y be the sample median and Z = max(Xi) be the sample maximum (a) Apply the usual formula for the density of an order statistic to show the density of Y is (b) Note that a beta random variable X has density f(x) = TaT(可 with mean μ = α/(a + β) and variance σ2 = αβ/((a +s+...
4. An exponential random variable X has p.d.f (x,07/», x>0, c.df. F(x:0) 1-exp(-x/8) for x > 0, and mean θ. A single observation of an exponential random variable X is used to test H0 : θ-2 against H1 : θ-5. The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. (a) What is the probability of committing a Type I error? (b) What is the probability of committing a Type...
Consider two pdfs f(C) and f, (x) which are defined as follows 1 0sxs1 f(x)=16 otherwise and i(x)- 0 otherwise Suppose that a single observation X is to be taken from f(x) where f(x) is either :G(x) or f(x). The following simple hypotheses are to be tested: Ho:(x)- (x) 4: f (x) = f(x) Determine a test procedure δ for which the value of α (6) + 23(6) is minimum. Report both the rejection region and the corresponding minimum value...
Let X1, X2,.,X10 be a sample of size 10 from an exponential distribution with the density function Sae -Xx f(x; A) otherwise 10 We reject Ho : ^ = 1 in favor of H : 1 = 2 if the observed value of Y = smaller than 6 (a) Find the probability of type 1 error for this test. (b) Find the probability of type 2 error for this test (c) Let y5 be the observed value of Y. Find...
Problem H5 Let X be a single observation (n-1) from the following distribution: f(rle)-o elsewhere NOTE:XBeta(0, 1) The following two hypotheses are being tested: 110 : e-2 vs Ha : ?-1. (a) Draw a graph of f(z | ?) when (i) Ho is true and when (ii) H. is true. Put both graphs on the same plot. Explain why a rejection region of the form (X<k) makes intuitive sense (b) Find k, so that the test has level a 0.05....
3. Let X, X,. X, be a random sample from a distribution with density f(x)- ) if 0 < x < θ 0 otherwise (a) (b) Determine the density of =max(X1,X2, ,Xn} Use the result of (a) to show that θ is a biased estimator of θ. Determine the bias, B(6) Calculate the mean-square error, MSE(6). (c)
MA2500/18 8. Let X be a random variable and let 'f(r; θ) be its PDF where θ is an unknown scalar parameter. We wish to test the simple null hypothesis Ho: 0 against the simple alternative Hi : θ-64. (a) Define the simple likelihood ratio test (SLRT) of Ho against H (b) Show that the SLRT is a most powerful test of Ho against H. (c) Let Xi, X2.... , X be a random sample of observations from the Poisson(e)...
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