**Problem Description:**1. The random variable \( z \) describes the observation of a physical phenomenon whose statistical description depends on one of two hypotheses. Conditioned on hypothesis \( H_1 \), the random variable \( z \) follows an Erlang distribution with parameters \( (3, 0.1) \). Conditioned on hypothesis \( H_0 \), the random variable \( z \) follows an Erlang distribution with parameters \( (1, 0.1) \). The prior probabilities are given as: \[ \Pr\{H_1\} = 0.65, \quad \Pr\{H_0\} = 0.35 \] Tasks: a) Develop the likelihood ratio to implement a MAP (Maximum A Posteriori) decision test in order to decide between the two hypotheses. b) Obtain the probability of error of the test. c) If \( z = 15.5 \), determine the decision result from the test.

The PDF of Erlang distribution is given by:

Answered: 1. The random variable z describes the…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Similar questions

1) Let X₁, X₂,..., X, be random sample of size n that distributed according to exponential distribution with mean a) Explain why Y() is not a pivotal quantity? (Note that Y() is the maximum order statistic). nY (1) b) Explain why Q= is a pivotal quantity? (Note that Y) is the 1st order 0 statistic). c) Based on the pivotal quantity, derive the 90% confidence interval for 0. (Identify values of a and b in the confidence interval).
Calculate the value of the unpooled test statistic t in each of the following situations. In each case, assume the null hypothesis is Ho: H, - H, = 0. (Ro (a) x, = 38, s, = 8, n, = 64; x2 = 36, s2 = 7, n, = 49 t = (b) The difference in sample means is 43, s.e.(x, - x2) = 27.
A biologist believes that there has been increase in the mean total suspended solids (TSS, mg/L) among Philippine rivers since its last study, where a mean 80 mg/L of TSS was observed. A random sample of 25 rivers was observed and the mean TSS was found to be 100 mg/L. Suppose it is of interest to test the biologist claim, the appropriate pair of null and alternative hypotheses is O Ho: µ-80, Ha: μ>80 O Ho: μ-100, Ha: µ>100 O Ho: X-bar = 80, Ha: X-bar > 80 O Ho: X-bar = 100, Ha: X-bar > 100
1.ln a simple model Y = Bo + B1X +u, X is endogenous, and Z is an instrumental variable that satisfy the relevance and exogeneity conditions. Show that the two stage least squares (TSLS) estimator is consistent. 2.In above model, what is the asymptotic distribution of TSLS? How can we do a). hypothesis testing of Bi and b). construct confidence interval for B,?
a) QUESTION 3 Suppose that Y₁, Y2, ..., Yn is a random sample from a normal distribution with parameter mean 0 and variance 0, say N(0,0). If Mr Masela and Mr Sepuru desire to use 5% level of significance to test the null hypothesis Ho: 0 = 1 against alternative hypothesis H₁:0 > 1. i) Show that the uniformly most powerful test of their hypothesis rejects Ho if Σ₁ yc, where c is a constant that solves the probability equation 0.05 = P₁Y≥ c|0 = 1). ii) What will be the value of c in part a) if n = 15.
18. Show that for type lII population - x/0 dp O
a) Suppose that Y₁, Y₂, ..., Y₁ is a random sample from a normal distribution with parameter mean 0 and variance e, say N(0,0). If Mr Masela and Mr Sepuru desire to use 5% level of significance to test the null hypothesis Ho: 0 = 1 against alternative hypothesis H₁:0 > 1. i) Show that the uniformly most powerful test of their hypothesis rejects H, if Σyc, where c is a constant that solves the probability equation 0.05=PY≥c|0 = 1). ii) What will be the value of c in part a) if n = 15. b) An academic Professor conducted a research aimed at exploring the experience and challenges faced by people who are diagnosed with dementia and their families. If the professor perform a hypothesis test to decide whether the mean age of all people diagnosis with initial occurrence of dementia is less than 55 years old. Based on the statement, explain, what each of the following would mean. i) Type I error. ii) Correct decision. iii) Type II error. c) Suppose that the results of carrying out the…

Recommended textbooks for you

MATLAB: An Introduction with Applications

Statistics

ISBN:

9781119256830

Author:

Amos Gilat

Publisher:

John Wiley & Sons Inc

Probability and Statistics for Engineering and th…

Statistics

ISBN:

9781305251809

Author:

Jay L. Devore

Publisher:

Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…

Statistics

ISBN:

9781305504912

Author:

Frederick J Gravetter, Larry B. Wallnau

Publisher:

Cengage Learning

MATLAB: An Introduction with Applications

Statistics

ISBN:

9781119256830

Author:

Amos Gilat

Publisher:

John Wiley & Sons Inc

Probability and Statistics for Engineering and th…

Statistics

ISBN:

9781305251809

Author:

Jay L. Devore

Publisher:

Cengage Learning

Statistics for The Behavioral Sciences (MindTap C…

Statistics

ISBN:

9781305504912

Author:

Frederick J Gravetter, Larry B. Wallnau

Publisher:

Cengage Learning

Elementary Statistics: Picturing the World (7th E…

Statistics

ISBN:

9780134683416

Author:

Ron Larson, Betsy Farber

Publisher:

PEARSON

The Basic Practice of Statistics

Statistics

ISBN:

9781319042578

Author:

David S. Moore, William I. Notz, Michael A. Fligner

Publisher:

W. H. Freeman

Introduction to the Practice of Statistics

Statistics

ISBN:

9781319013387

Author:

David S. Moore, George P. McCabe, Bruce A. Craig

Publisher:

W. H. Freeman

SEE MORE TEXTBOOKS