Problem 3. [20 points] Let X1,X2,,Xn be a sequence of i.i.d. r.v.s. with density fXi(x;)={1,0,if0x
otherwise where >0 is an unknown parameter. - a) [6 points] Find the method of moments
estimator ^ for (using the first moment). - b) [7 points] Assuming n is large enough for so that we
can use the same test as if the data was normally distributed, find the 95% confidence interval for
, based on ^. In particular, compute it if n=100 observations have the sample mean X=0.9 and the
sample variance Sn2=0.3. (Note: If you have not been able to solve part a), you may assume that
^=a+bX for some a and b>0 to solve part b).) - c) [7 points] Suppose we know that cannot be
larger than 2. How large does the sample size n have to be in order for ^ to have at least 80%
chance (approximately) of lying between 0.1 and +0.1 ? Here, you don't know the sample
variance, but you can replace it by the (conservative) estimate of the true variance 2=2(), using the
value =2. For this, you first have to compute the expression for the variance. ('Conservative' here
means using the upper bound for the variance.).
Problem 3 20 points Let X1X2Xn be a sequence of iid.pdf
1. Problem 3. [20 points] Let X1,X2,,Xn be a sequence of i.i.d. r.v.s. with density fXi(x;)={1,0,if0x
otherwise where >0 is an unknown parameter. - a) [6 points] Find the method of moments
estimator ^ for (using the first moment). - b) [7 points] Assuming n is large enough for so that we
can use the same test as if the data was normally distributed, find the 95% confidence interval for
, based on ^. In particular, compute it if n=100 observations have the sample mean X=0.9 and the
sample variance Sn2=0.3. (Note: If you have not been able to solve part a), you may assume that
^=a+bX for some a and b>0 to solve part b).) - c) [7 points] Suppose we know that cannot be
larger than 2. How large does the sample size n have to be in order for ^ to have at least 80%
chance (approximately) of lying between 0.1 and +0.1 ? Here, you don't know the sample
variance, but you can replace it by the (conservative) estimate of the true variance 2=2(), using the
value =2. For this, you first have to compute the expression for the variance. ('Conservative' here
means using the upper bound for the variance.)
![Problem 3. [20 points] Let X1,X2,,Xn be a sequence of i.i.d. r.v.s. with density fXi(x;)={1,0,if0x
otherwise where >0 is an unknown parameter. - a) [6 points] Find the method of moments
estimator ^ for (using the first moment). - b) [7 points] Assuming n is large enough for so that we
can use the same test as if the data was normally distributed, find the 95% confidence interval for
, based on ^. In particular, compute it if n=100 observations have the sample mean X=0.9 and the
sample variance Sn2=0.3. (Note: If you have not been able to solve part a), you may assume that
^=a+bX for some a and b>0 to solve part b).) - c) [7 points] Suppose we know that cannot be
larger than 2. How large does the sample size n have to be in order for ^ to have at least 80%
chance (approximately) of lying between 0.1 and +0.1 ? Here, you don't know the sample
variance, but you can replace it by the (conservative) estimate of the true variance 2=2(), using the
value =2. For this, you first have to compute the expression for the variance. ('Conservative' here
means using the upper bound for the variance.)](https://image.slidesharecdn.com/problem320pointsletx1x2xnbeasequenceofiid-230320092239-c38e83fa/85/Problem-3-20-points-Let-X1X2Xn-be-a-sequence-of-iid-pdf-1-320.jpg)
