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ramj
Questions (12)
Consider an i.i.d. sample X1,…,Xn∼Poiss(λ) for λ>0 .
Starting from the Central Limit Theorem, find a confidence interval
0 answers
1,380 views
Suppose that the rejection region of a test ψ has the form R={Xn:Xn>c} . Find the smallest c such that ψ has level α .
(If
2 answers
1,397 views
Let X be a single (i.e. n=1 ) Gaussian random variable with unknown mean μ and variance 1 . Consider the following hypotheses:
H
1 answer
1,658 views
Let 𝑋1,…,𝑋𝑛 be i.i.d. Bernoulli random variables with unknown parameter 𝑝∈(0,1) . Suppose we want to test
𝐻0:�
1 answer
2,066 views
a)𝑋1,…,𝑋𝑛∼𝑖.𝑖.𝑑.𝖯𝗈𝗂𝗌𝗌(𝜆) for some unknown 𝜆>0 ;
𝐻0:𝜆=𝜆0 v.s.
1 answer
1,561 views
The National Assessment of Educational Progress tested a simple random sample of 1000 thirteen year old students in both 2004
1 answer
1,830 views
Suppose that the rejection region of a test 𝜓 has the form 𝑅={𝑋⎯⎯⎯⎯⎯𝑛:𝑋⎯⎯⎯⎯⎯𝑛>𝑐} . Find
0 answers
566 views
The Gamma distribution Gamma(𝛼,𝛽) with paramters 𝛼>0 , and 𝛽>0 is defined by the density
𝑓𝛼,𝛽(𝑥)=𝛽𝛼
6 answers
2,753 views
Argue that the proposed estimators 𝜆ˆ and 𝜆˜ below are both consistent and asymptotically normal. Then, give their
1 answer
2,005 views
Let 𝑋1,…,𝑋𝑛 be i.i.d. random variables with distribution (𝜃,𝜃) , for some unknown parameter 𝜃>0 .
Find an
2 answers
3,013 views
Let X1,…,Xn be i.i.d. Poisson random variables with parameter λ>0 and denote by X¯¯¯¯n their empirical average,
X¯¯¯¯n
2 answers
2,300 views
A random variable X is generated as follows. We flip a coin. With probability p , the result is Heads, and then X is generated
1 answer
1,494 views
Answers (4)
Suppose that the test ψ has level α=0.05 . What is the power of ψ ? (If applicable, type abs(x) for |x| , Phi(x) for Φ(x)=P(Z≤x) where Z∼N(0,1) , and q(alpha) for qα , the 1−α quantile of a standard normal variable, e.g. enter q(0.01) for q0.01
Using the previous part, find confidence intervals for 𝛼 with asymptotic level 90% using both the “solving" and the “plug-in" methods. Use 𝑛=25 , and 𝑋⎯⎯⎯⎯⎯𝑛=4.5 . (Enter your answers accurate to 2 decimal places. Use the Gaussian
clarification : Let 𝑋1,…,𝑋𝑛 be i.i.d. random variables with distribution N(𝜃,𝜃) , for some unknown parameter 𝜃>0 .
Secondly, express P(|Z|≤t) in terms of Φ(r)=P(Z≤r) for t>0 . Write Phi(t) (with capital P) for Φ(t) .
