# Advance Probability Problems

**Paper, Order, or Assignment Requirements**

it is an urgent assignment related to probability and statistics

- Suppose you have a random sample of size n from the double-exponential (or Laplace) distribution with densityon the whole real line. Find a sufficient statistic for the parameter. Is this sufficient statistic complete? Find the method-of-moments estimator of the parameter. Find the MLE of the parameter. Are any of these two estimators unbiased? Find an unbiased estimator of the parameter. Can you derive the RCLB for the variance of an unbiased estimator of?
- In the above problem, derive the Neyman-Pearson most powerful test for testing. Can you find the exact critical region for this test if the alpha level is 0.05? Comment on how this test needs to be modified for testing.
- Suppose you have a random sample of size n from an Inverse Gaussian distributions with parameters, whose density is on x > 0. Find jointly sufficient statistics for the two parameters. Now suppose that=50. Can you find a complete sufficient statistic for the other parameter? Next, suppose thatis unknown but. Can you find a complete sufficient statistic for?
- Suppose you have a random sample of size n from Normal(), with both parameters unknown. If b is the 90
^{th}percentile of this distribution (i.e. if P(X b) = 0.9 where X has this normal distribution),find the MLE and the MVUE of b. Next, if c is a given constant, find the MLE and the MVUE of P(X c). - For a random sample of size 1 from Uniform(), find a complete, sufficient statistic for the parameter (you have to show that it is complete). Next, show that this complete, sufficient statistic is independent of Y= sign(X) [
**Note**: Y = sign(X) is the function of X that takes the value 1 if X >0 and the value -1 if X < 0].

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