# probability theory and statistics 15

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1. The mean square error of the sample mean of *n *observations (used as an estimator for the true mean)

is:

The square root of this quantity, *e*, is often referred to as the *standard error *of the estimate.

*e*=*E*[(*X*âˆ’*Î¼**X*)2]=*Var*(*X*)= *Var*(*X*) âˆš*n *

- (a) For the case that
*X*is a Gaussian random variable, confirm that the sample mean will be within one standard error of*Î¼**X*with probability 0.68. - (b) Suppose we perform a sequence of
*n*independent Bernoulli(*p*) trials. How many trials are required to guarantee that*X*, used to estimate*p*, has standard error â‰¤ 0.05 ?

2. *X *and *Y *are joint random variables. In order to estimate the covariance of *X *and *Y *, we make *n *joint measurements independently; the results are (*X*1,*Y*1),(*X*2,*Y*2),…(*X**n*,*Y**n*). Note that *X**i *and *Y**j *are independent random variables if *i *Ì¸= *j *(because they come from different measurements), but they are not independent if *i *= *j *(because they come from the same measurement.) Consider the following estimator for the covariance:

*C*(*X*,*Y*)= 1âˆ‘(*X**i*âˆ’*X*)(*Y**i*âˆ’*Y*) *n**i *

Calculate the bias of *C*, which is *E*[*C*]âˆ’*Cov*(*X*,*Y*), the expected difference between the estimated covariance and the true covariance. Is *C *unbiased? If it is biased, can you fix it? *A useful result: if P and Q are independent random variables, E*[*PQ*] = *E*[*P*]*E*[*Q*]*. *

- Given the following i.i.d. observations of a random variable
*Y*: 15.2, 12.0, 14.0, 13.4, 15.0, 12.5- (a) Calculate the sample mean, sample variance, and sample standard deviation.
- (b) Calculate an Î± = 0.05 confidence interval for the mean, using the Central Limit Theorem approximation.
- (c) Calculate an Î± = 0.05 confidence interval for the mean, using Studentâ€™s
*t*-distribution. (A*t*-table is posted on the eLearning website.) - (d) Estimate how many measurements you would need to make in order to get a confidence interval for the mean that is Â±1.0 wide. Answer this using both the Central Limit Theorem and the

*t*-distribution. - (e) Calculate an Î± = 0.05 confidence interval for the variance, using the Ï‡2 distribution. (A Ï‡2 table is posted on the eLearning website.)

- Suppose that you are testing a drug which is supposed to prevent strokes. In the general population, 80-year-old people have a â€œfirst-stroke annual rateâ€ of 1.4%; that is,
*P*= 0.014 that a randomly selected 80-year old who hasnâ€™t had a stroke before will have one in the coming year. You recruit 10,000 such 80-year-olds to your study. You give half the drug, and half a placebo (an inactive substitute pill, which shouldnâ€™t have an effect.)- (a) Calculate the expected number of strokes in the placebo group, and estimate a Î± = 0.05 confidence interval for this quantity.
- (b) Suppose that, after a year, 61 people in the â€œdrugâ€ group have had strokes, and 73 people in the â€œplaceboâ€ group have had strokes. Do you think the drug actually works? Why, or why not? Justify your answer by estimating the probability of a stroke in each group, along with an

Î± = 0.05 confidence interval.