Tag Archives: probability

DSL for Generative Models (51/365)

The backlog becomes longer. I’ve changed jobs two weeks ago and it has upset my routine. No matter. Here we go again. I want to deviate from my problem solving mode for a while and use up a few posts … Continue reading

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Random Variables – Problem (50/365)

Moving on to the next chapter “Random Variables – I”, take a look at the following problem. Show that the random variable is continuous if and only if for all . (Forward direction) Suppose is a continuous random variable, then … Continue reading

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Measurable Spaces – Problem (49/365)

Let be the Lebesque-Stieltjes measure generated by a continuous function. Show that if the set is at most countable, then . A Lebesque-Stieltjes measure is a countably additive measure and is given by a generalized distribution function such that that … Continue reading

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Measurable Spaces – Problem (48/365)

Another question on distribution functions. Show that each of the functions is continuous on the right but is not a distribution function in . Take the first function. To show that it is continuous on the right, let and let … Continue reading

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Measurable Spaces – Problem (47/365)

The previous post involved distribution functions over the real numbers but it’s also possible to have distribution functions over . A problem asks to show that if we have the distribution function And a difference function Then show that Just … Continue reading

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Measurable Spaces – Problem (46/365)

The rest of the chapter on -algebras goes through the construction of various other measurable spaces such as those on the space of 1) continuous functions, 2) functions continuous on the right, and 3) direct products of measurable spaces. The … Continue reading

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Measurable Spaces – Problem (45/365)

Show that the following is not a Borel set in (this is the -algebra of functions over the domain unlike the previous onces we looked at where the domain was over the natural numbers). If is a Borel set, then … Continue reading

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