The heights1,2,...,10of a random sample of 10 people are i.i.d. normal random(,2)variables. Assume the heights are measured in inches.

4. Size of Error, Part I

Data scientists often have to estimate the size of a random error. A simple model for a random error is that it is distributed uniformly on the interval(?,?)for some constant?>0

If?is unknown, data scientists might have to estimate it. Suppose they have just one data point? distributed uniformly on(?,?). Let's see how they can use it to estimate?

a)The first step is to simplify the problem. Let?=|?|. What are the possible values ofjQuery22405483736774652446_1606196206402 For each possible value? find?(?>?)and hence identify the distribution of?

b)To estimate? suppose the data scientists use the estimator?1=?. Find the bias of?1 and also the variance of?1.

c)Use?1to construct an unbiased estimator of?. Call this new estimator?2. The good news is that it is unbiased by your construction. Now find???(?2)and compare it with???(?1)in Partb.

5. Size of Error, Part II

Now let?1,?2,...,jQuery22406114161506566345_1606195888673 X1,X2,...,Xnbe i.i.d. uniform on(0,?) (0,). As in the previous exercise, the goal is to use the data to estimate?.

a)Let?be the sample mean. Use?to construct an unbiased estimator?of the parameter?, and find??? ? (?)

b)Let?=max(?1,?2,...,??) M=max(X1,X2,...,Xn). Find? ?(?)by finding first the cdf, then the density, and then the expectation.

c)Explain your answer for??(?) by a symmetry argument. [Draw the interval(0,?). Take?=3 or some other small integer, and throw downnpoints on the interval. Stare at the gaps created. How many are there? How many between0andM?]

d)Find the bias of? and say what happens to the bias as the sample size gets large.

e)FindVar(M). For this you do need the density you found in Partb.

f)If the sample is large, which of the two estimatorsMand?1 would you rather use, and why? Make a strong case for your choice. You are welcome to support your answer by algebra/calculus or numerical computation.