2. 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 \( z \) is Gaussian \((200, 10)\), while conditioned on hypothesis \( H_0 \), the random variable \( z \) is Gaussian \((150, 10)\). If \( \text{Pr}\{H_1\} = 0.1 \), and \( \text{Pr}\{H_0\} = 0.9 \):a) Develop the likelihood ratio to implement a MAP decision test in order to decide between the two hypotheses.b) Obtain the probability of error of the test.c) If \( z = 165 \), what decision results from the test?

The statistical problem involves the observation of physical phenomena described by random variable…

Answered: 2. The random variable z describes the…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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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,?
4- A gambler plays roulette game and he is insisting to play all his money to black (A roulette wheel has 38 slots: 18 are black, 18 are red, and 2 are green). a) Try to calculate required number of spin for average(µ) loss corresponds to the certain amount of his initial money (M) completely. Around this average loss, what is the confidence interval (95% or u±20 ) of the money (the variance (O? ) of the process should be calculated as well)? b)RV x is money (won (n times) or lost(N-n times) amount where N is number of spin, e.g. 25, 50, 100, 200, 400, 500,..1500...) try to establish a distribution and all moments of this RV. Try to explain whether his decision is sustainable or not. Try to find a winning strategy. Plot the probability density function for various N values.
State which test (z-test, t-test, ANOVA, χ 2 test) is most suitable for each of the following cases: (i) In an election, voters may support the Democrat candidate, Republican candidate, or Independent candidate. A researcher would like to conduct a poll to test if this year the supportive rates of the three candidates follow the same supportive rates as in the last election, which was 60% : 25% : 15%. (ii) A vaccine is developed for preventing the infection of a disease. An experiment is developed to test its effectiveness. 300 volunteers are divided into two groups randomly. One group of participants will take the vaccine and the other group of participants will not take the vaccine but only vitamin. The proportion of participants in the two groups being infected by the disease will be compared. (iii) A researcher wants to conduct a test to verify if children’s outside school learning experiences would be affected by the families’ income level. Family income will be classified as high…
A random sample is selected from a normal population with a mean u = 30 and standard deviation sima = 8 . After a treatment is administered to the individuals in the sample, the sample mean is found to be M =33. a) if the sample consist of n=16 scores is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two tailed test with alpha spaceequals space 0.05. b) if a sample consists of n= 64 scores, is the sample mean sufficient to conclude that the treatment has a significant effect? Use a two tailed with alpha space equals space 0.05. c) comparing your answers for parts a and b explain how the size of the sample influences the outcome of the hypothesis test?

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