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Show that for some topological spaces (X, S) and (Y, T ), there is a closed set D in X × Y with product topology which is not in any product σ-algebra A ⊗ B, for example if A and B are the Borel σ-algebras for the given topologies. Hint: Let X = Y be a set with cardinality greater than c, for example, the set 2I of all subsets of I := [0, 1] (Theorem 1.4.2). Let D be the diagonal {(x, x ): x ∈ X }. Show that for each C ∈ A ⊗ B, there are sequences { An } ⊂ A and {Bn } ⊂ B such that C is in the σ-algebra generated by { An × Bn }n ≥ 1. For each n, let x =n u mean that x ∈ An if and only if u ∈ An . Define a relation x ≡ u iff for all n, x =n u. Show that this is an equivalence relation which has at most c different equivalence classes, and for any x, y, and u, if x ≡ u, then (x, y) ∈ C if and only if (u, y) ∈ C . For C = D and y = x , find a contradiction.
a distribution has a standard deviation of s 12. find the z- score for each of the following locations in the
Create a dot plot and a box plot to represent this data. If we create a sample by picking every 3rd data from the population, how would your answer to part a change? Show your work
A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black.
Using a .05 level of significance, test whether the sample results indicate that the manufacturing process is functioning properly.
The mean area of several thousand apartments in a new development is advertised to be 1250 sq. ft. An engineer measured a random sample of apartments:
Packets filled consecutively by the machine. Calculate the limits within which the weight of the packets should lie assuming that the machine is not been classified as faulty.
The mean SAT verbal score is 412, with a standard deviation of 90. Use the Empirical Rule to determine what percent of the scores lie between 412 and 592. (Assume the data set has a bell-shaped distribution.)
What is the probability of winning if the rules are changed so that in addition to selecting the correct five numbers, you must now select them in the same order they are drawn?
question hicks law predicts how long it will take a user to pick out a given element on a website page from among all
What does this data tell us by itself before we do anything with it such as doing hypothesis testing or comparing it to other data? How can this be used by the M&M company?
Decileter and standard deviation of 41 mg/dl. in an srs of 100 men what is the probability that x bar takes a value between 185 and 191 mg/dl?
suppose you work for an automotive engineering group that is responsible for modeling the length of life of brake
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