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1. Let (S, d) be a complete separable metric space with S non-empty. Sup- pose S has no isolated points, that is, for each x ∈ S, x is in the closure of S\{x }. Prove that there exists a Borel measure µ on S with µ(S) = 1 and µ({x }) = 0 for all x ∈ S. Hint: Define a 1-1 Borel measurable function f from [0, 1] into S and let µ = λ ? f -1.
2. Show that there is a (non-Hausdorff) topology on a set X of two points for which the Baire and Borel σ-algebras are different.
3. A collection L of subsets of a set X will be called a lattice iff /0 ∈ L, X ∈ L, and for any A and B in L, A ∪ B ∈ L and A ∩ B ∈ L. Show that then the collection D of all sets A\B for A and B in L is a semiring (§3.2). Then show that the algebra A generated by (smallest algebra including) a lattice L is the collection of all finite unions l1≤i ≤n Ai \Bi for Ai and Bi in L, where the sets Ai \Bi can be taken to be disjoint for different i. (In any topological space, the collection of all open sets forms a lattice, as does the collection of all closed sets.)
a doctors office staff studied the waiting times for patients who arrive at the office with a request for emergency
Prepare an ANOVA table and test the null hypothesis of no linear relationship between the two variables.
a sample of 40 los angeles pharmacies sells the tamiflu 75mg prescription drug for about 137 for a moth supply s 30.2
When considering the symmetric approximation and Monte Carlo risk approaches. Both make the assumption that the total cost distribution is normal.
1. in a hospitals shipment of 3500 insulin syringes 14 were unusable due to defects. is this sufficient evidence to
The following confidence interval is obtained for a population proportion, p:(0.458, 0.490). Use these confidence interval limits to find the point estimate, p-hat.
Do the data provide significant support for any preferences among the three brands? Test At the .05 level of significance.
The following data represents the pulse rates of ten randomly selected females after stepping up and down on a 6-inch platform for 3 minutes. Pulse is measured in beats per minute. Compute the sample mean pulse.
The assembly time for a product is uniformly distributed between 6 to 10 minutes. The probability of assembling the product between 7 to 9 minutes is
heights of men on a baseball team have a bell-shaped distribution with a mean of 169 cm and a standard deviation of 5
If a random sample of 10 people found that 9 were pro-life (i.e., 90%), while another random sample of 1000 people from the same population found that 550 were pro-life (i.e., 55%), which would you find more significant? Why?
What deccision should be made about the null hypothesis, and what decision should be made about the effect of the program.
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