Problem 4 A confidence interval for uniform distributions Let X1,..., X, be i.i.d. uniform random variables in [0, 0), for some 0 > 0. Denote by M, = max X;. i=1,..,n 1. Prove that Mn converges in probability to 0.
Problem 4 A confidence interval for uniform distributions Let X1,..., X, be i.i.d. uniform random variables in [0, 0), for some 0 > 0. Denote by M, = max X;. i=1,..,n 1. Prove that Mn converges in probability to 0.
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![Problem 4
A confidence interval for uniform distributions
Let X1,..., X, be i.i.d. uniform random variables in [0, 0], for some 0 > 0. Denote by
max X,.
i=1,...,n
1. Prove that Mn converges in probability to 0.
2. Compute the cumulative distribution function of n(1 – Mn/0) and prove that
n(1 – Mn/0) converges in distribution to an exponential random variable with
parameter 1.
3. Use the previous question to find an interval I of the form I = [Mn, Mn +c], that
does not depend on 0 and such that
P[I > 0) -
→ .95,
n → 0.
0 > M„]
4. Is M, unbiased?
[Hint:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd33fd28-1035-47d9-bf5b-a7111879790a%2Ff84a7635-25d0-4868-83c6-3d8156daca45%2Fcm9w9gj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 4
A confidence interval for uniform distributions
Let X1,..., X, be i.i.d. uniform random variables in [0, 0], for some 0 > 0. Denote by
max X,.
i=1,...,n
1. Prove that Mn converges in probability to 0.
2. Compute the cumulative distribution function of n(1 – Mn/0) and prove that
n(1 – Mn/0) converges in distribution to an exponential random variable with
parameter 1.
3. Use the previous question to find an interval I of the form I = [Mn, Mn +c], that
does not depend on 0 and such that
P[I > 0) -
→ .95,
n → 0.
0 > M„]
4. Is M, unbiased?
[Hint:
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