# probability

Let X be a continuous random variable. We know that it takes values between 0 and 3, but we do not know its distribution or its mean and variance. We are interested in estimating the mean of X, which we denote by h. We will use 1.5 as a conservative value (upper bound) for the standard deviation of X. To estimate h, we take n i.i.d. samples X1,X2,…,Xn, which all have the same distribution as X, and compute the sample mean

H=1n∑i=1nXi.
Express your answers for this part in terms of h and n using standard notation.

Given the available information, the smallest upper bound for var(H) is: - unanswered
Calculate the smallest possible positive value of n such that the standard deviation of H is guaranteed to be at most 0.01.

This minimum value of n is: - unanswered
We would like to be at least 99% sure that our estimate is within 0.05 of the true mean h. Using the Chebyshev inequality, calculate the minimum value of n that will achieve this.

This minimum value of n is: - unanswered
Assume that X is uniformly distributed on [0,3]. Using the Central Limit Theorem, identify the most appropriate expression for a 95% confidence interval for h.

- unanswered [H−1.96n−−√,H+1.96n−−√] [H−1.96⋅3−−−−−−√4n−−√,H+1.96⋅3−−−−−−√4n−−√] [H−1.963√4n−−√,H+1.963√4n−−√] [H−1.96⋅34n−−√,H+1.96⋅34n−−√

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1. 5. H-1.96*(3^0.5)/((4*n)^0.5),
H+1.96*(3^0.5)/((4*n)^0.5)

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2. does someone have an answer for this question? i am stuck

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4. Help is much appreciated, please!

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