(Hypothesis Testing: Uniform and Uniform) Consider the binary hypothesis testing problem in which the hypotheses H=0...
(Hypothesis Testing: Uniform and Uniform) Consider the binary hypothesis testing problem in which the hypotheses H=0 and H=1 occur with probability PH(0) and PH(1)=1- PH(0), respectively. The observation Y is a sequence of zeros and ones of length 2k, where k is a fixed integer. When H=0, each component of Y is 0 or a 1 with probability ½ and components are independent. When H=1, Y is chosen uniformly at random from the set of all sequences of length 2k that have an equal number of ones and zeros. There are such sequences.
(a) What is PY|H(y|0)? What is PY|H(y|1)?
(b) Find the maximum likelihood decision rule. What is the single number you need to know about yto implement this decision rule?
(c) Find the decision rule that minimizes the error probability.
(d) Are there values of PH(0) and PH(1) such that the decision rule that minimizes the error probability always decides for only one of the alternatives? If yes, what are these values, and what is the decision?
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(Hypothesis Testing: Uniform and Uniform) Consider the binary
hypothesis testing problem in which the hypotheses H=0...
Problem 6 Hypothesis Testing: Uniform and Uniform) Consider a binary hypothesis testing problem i...
Problem 6 Hypothesis Testing: Uniform and Uniform) Consider a binary hypothesis testing problem in which the hypotheses H 0 and H 1 occur with probability PH (0) and PH (1) 1-PH (0), respectively. The observation Y is a sequence of zeros and ones of length 2k, where k is a fixed integer. When H 0, each component of Y is 0 or a 1 with probability and components are independent, when H = 1, Y is chosen uniformly at random...
decide Tthe problern is easy? Problem 2 (The "Weather frog") Let us assume that a "weather...
decide Tthe problern is easy? Problem 2 (The "Weather frog") Let us assume that a "weather frog" bases his forecast for tomorrow's weather entirely on today's air pressure. Determining a weather forecast is a hypothesis testing problem. For simplicity, let us assume that the weather frog only needs to tell us if the forecast for tomorrow's weather is "sunshine" or "rain". Hence we are dealing with binary hypothesis testing. Let H 0 mean "sunshine" and H 1 mean "rain". We...
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS θ for the purpose of testing Ho : θ-2 against H, : θ < 2 at α : 0.10 level of significance. Suppose the decision rule is to be...
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS θ for the purpose of testing Ho : θ-2 against H, : θ < 2 at α : 0.10 level of significance. Suppose the decision rule is to be based on Xmax, the largest order statistic. What would be the probability of committing a Type II error when θ 1.7.
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS...
1. In hypothesis testing, the hypothesis that is assumed to be true for the purpose of...
1. In hypothesis testing, the hypothesis that is assumed to be true for the purpose of testing is called the hypothesis 2. (Circle the correct response) In hypothesis testing, critical values used to make a rejection decision regarding the null hypothesis are determined by the nature of the hypothesis test (two-tail vs. one-tail) and the d. significance level a. sample size b. population parameter c. target value 3. (Circle the correct response) In the process of hypothesis testing, the test...
Consider a binary erasure channel, in which the input X ∼ Bernoulli ? (1 ?, 3)...
Consider a binary erasure channel, in which the input X ∼
Bernoulli ? (1 ?,
3)
and the output Y ∈ {0, e, 1} where the symbol e denotes an
erasure event (e appears when the channel is too “bad”). The
conditional distribution of Y given X is as follows:
pY |X (0|0) = 0.9, pY |X (e|0) = 0.1, pY |X (1|1) = 0.8, pY |X
(e|1) = 0.2.
Given that an erased symbol has been observed, i.e., Y...
Suppose that you are testing the hypotheses Upper H 0: p=0.38 vs. Upper H Subscript Upper...
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You are testing the null hypothesis that there is no linear relationship between two variables, X...
You are testing the null hypothesis that there is no linear relationship between two variables, X and Y. You are given the following regression results, where the sample size is 10., Coefficients Standard Error Intercept -1.2 1 X 2 2 a) What is the value of the t test statistic? b) At the α = 0.05 level of significance, what are the critical values? c) Based on your answer to (a) and (b), what statistical decision should you make?
Consider this testing situation. A box contains 16 chips (with some mixture of red and black...
Consider this testing situation. A box contains 16 chips (with
some mixture of red and black chips). Suppose we have the following
hypotheses:
HO: The box contains R=8 red and
B=8 black chips.
HA: The box contains some other
mixture of red and black chips.
We randomly select 5 chips simultaneously from the box without
replacement. Our Test Statistic is the Y = # of Black chips found
in the sample.
Suppose we use the following decision rule:...
1. Suppose that you are testing the following hypotheses: ?0: ? = 10 and ?1: ?...
1. Suppose that you are testing the following hypotheses: ?0: ? = 10 and ?1: ? > 10. If the null hypothesis is rejected at the 1% level of significance, what statement can you make about the confidence interval on the mean? a.) The lower bound of a 95% one-sided confidence interval on the mean exceeds 10 b.) 9 ≤ ? ≤ 13 c.) No statement can be made d.) The lower bound of a 95% one-sided confidence interval on...
MATLAB Notes: B) Find likelihood for each sequence 1-9 for each hypothesis. First hypothesis with the...
MATLAB
Notes: B)
Find likelihood for each sequence 1-9 for each
hypothesis.
First hypothesis with the fair coin (theta value = 0.5): what
is the probability that we can get each sequence, 1 through
9.
Second hypothesis with weighted coin(don’t know theta value):
what is likelihood of getting sequence 1 through 9. Using equation
to find all possible hundred theta values. For each theta value we
need to compute likelihood.
Then we sum them up across all possible theta values....
Problem 6 Hypothesis Testing: Uniform and Uniform) Consider a binary hypothesis testing problem in which the hypotheses H 0 and H 1 occur with probability PH (0) and PH (1) 1-PH (0), respectively. The observation Y is a sequence of zeros and ones of length 2k, where k is a fixed integer. When H 0, each component of Y is 0 or a 1 with probability and components are independent, when H = 1, Y is chosen uniformly at random...
decide Tthe problern is easy? Problem 2 (The "Weather frog") Let us assume that a "weather frog" bases his forecast for tomorrow's weather entirely on today's air pressure. Determining a weather forecast is a hypothesis testing problem. For simplicity, let us assume that the weather frog only needs to tell us if the forecast for tomorrow's weather is "sunshine" or "rain". Hence we are dealing with binary hypothesis testing. Let H 0 mean "sunshine" and H 1 mean "rain". We...
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS θ for the purpose of testing Ho : θ-2 against H, : θ < 2 at α : 0.10 level of significance. Suppose the decision rule is to be based on Xmax, the largest order statistic. What would be the probability of committing a Type II error when θ 1.7.
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS...
1. In hypothesis testing, the hypothesis that is assumed to be true for the purpose of testing is called the hypothesis 2. (Circle the correct response) In hypothesis testing, critical values used to make a rejection decision regarding the null hypothesis are determined by the nature of the hypothesis test (two-tail vs. one-tail) and the d. significance level a. sample size b. population parameter c. target value 3. (Circle the correct response) In the process of hypothesis testing, the test...
Consider a binary erasure channel, in which the input X ∼
Bernoulli ? (1 ?,
3)
and the output Y ∈ {0, e, 1} where the symbol e denotes an
erasure event (e appears when the channel is too “bad”). The
conditional distribution of Y given X is as follows:
pY |X (0|0) = 0.9, pY |X (e|0) = 0.1, pY |X (1|1) = 0.8, pY |X
(e|1) = 0.2.
Given that an erased symbol has been observed, i.e., Y...
Consider this testing situation. A box contains 16 chips (with
some mixture of red and black chips). Suppose we have the following
hypotheses:
HO: The box contains R=8 red and
B=8 black chips.
HA: The box contains some other
mixture of red and black chips.
We randomly select 5 chips simultaneously from the box without
replacement. Our Test Statistic is the Y = # of Black chips found
in the sample.
Suppose we use the following decision rule:...
MATLAB
Notes: B)
Find likelihood for each sequence 1-9 for each
hypothesis.
First hypothesis with the fair coin (theta value = 0.5): what
is the probability that we can get each sequence, 1 through
9.
Second hypothesis with weighted coin(don’t know theta value):
what is likelihood of getting sequence 1 through 9. Using equation
to find all possible hundred theta values. For each theta value we
need to compute likelihood.
Then we sum them up across all possible theta values....
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