Chapter 5: Q5SE (page 344)
Consider again the two tests A and B described in Exercise2. If a student is chosen at random, and her scoreon test B is 100, what predicted value of her score on test A has the smallest M.S.E., and what is the value of thisminimum M.S.E.?
The M.S.E value is 5.76
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Every Sunday morning, two children, Craig and Jill, independently try to launch their model airplanes. On each Sunday, Craig has a probability \(\frac{{\bf{1}}}{{\bf{3}}}\) of a successful launch, and Jill has the probability \(\frac{{\bf{1}}}{{\bf{5}}}\) of a successful launch. Determine the expected number of Sundays required until at least one of the two children has a successful launch.
Suppose that 16 percent of the students in a certain high school are freshmen, 14 percent are sophomores, 38 percent are juniors, and 32 percent are seniors. If 15 students are selected at random from the school, what is the
The probability that at least eight will be either freshmen or sophomores?
Let \({{\bf{X}}_{{\bf{1,}}}}{\bf{ }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\)be i.i.d. random variables having thenormal distribution with mean \({\bf{\mu }}\) and variance\({{\bf{\sigma }}^{\bf{2}}}\). Define\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}_{\bf{i}}}} \) , the sample mean. In this problem, weshall find the conditional distribution of each \({{\bf{X}}_{\bf{i}}}\)given\(\overline {{{\bf{X}}_{\bf{n}}}} \).
a.Show that \({{\bf{X}}_{\bf{i}}}\)and\(\overline {{{\bf{X}}_{\bf{n}}}} \) have the bivariate normal distributionwith both means \({\bf{\mu }}\), variances\({{\bf{\sigma }}^{\bf{2}}}{\rm{ }}{\bf{and}}\,\,\frac{{{{\bf{\sigma }}^{\bf{2}}}}}{{\bf{n}}}\),and correlation\(\frac{{\bf{1}}}{{\sqrt {\bf{n}} }}\).
Hint: Let\({\bf{Y = }}\sum\limits_{{\bf{j}} \ne {\bf{i}}} {{{\bf{X}}_{\bf{j}}}} \).
Nowshow that Y and \({{\bf{X}}_{\bf{i}}}\) are independent normals and \({{\bf{X}}_{\bf{n}}}\)and \({{\bf{X}}_{\bf{i}}}\) are linear combinations of Y and \({{\bf{X}}_{\bf{i}}}\) .
b.Show that the conditional distribution of \({{\bf{X}}_{\bf{i}}}\) given\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\overline {{{\bf{x}}_{\bf{n}}}} \)\(\) is normal with mean \(\overline {{{\bf{x}}_{\bf{n}}}} \) and variance \({{\bf{\sigma }}^{\bf{2}}}\left( {{\bf{1 - }}\frac{{\bf{1}}}{{\bf{n}}}} \right)\).
In Exercise 5, let \({{\bf{X}}_{\bf{3}}}\) denote the number of juniors in the random sample of 15 students and the number of seniors in the sample. Find the value of \({\bf{E}}\left( {{{\bf{X}}_{\bf{3}}} - {{\bf{X}}_{\bf{4}}}} \right)\) and the value of \({\bf{Var}}\left( {{{\bf{X}}_{\bf{3}}} - {{\bf{X}}_{\bf{4}}}} \right)\)
Suppose thatX1andX2are independent random variables, thatX1has the binomial distribution with parametersn1 andp, and thatX2has the binomial distributionwith parametersn2andp, wherepis the same for bothX1andX2. For each fixed value ofk (k=1,2,…, n1+n2),prove that the conditional distribution ofX1given thatX1+X2=kis hypergeometric with parametersn1,n2,andk.
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