Recent content by Edwinkumar

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    Undergrad Having problems with PMF, CDF and PMF.

    Usually in the undergraduate level pmf associated with a discrete random variable and pdf is with continuous variable. But both can be used interchangeably ( I think ).
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    Graduate Is marginal constraints equivalent to linear constraints?

    Could someone answer this?
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    Graduate Is marginal constraints equivalent to linear constraints?

    If I have a set of Probability distributions on a product space with marginal constraints, is there any way to (how to) express the same as a linear family of PD's ( i.e. all P s.t. E_P[ f_i] =a_i for some f_i, a_i )
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    Graduate Measure theoretically equivalent

    I couldn't find this terminology in books. When I was reading a paper I-divergence geometry of distributions by I.CSISZAR, I came across this term. Initially I thought it could be mutually absolutely continuous; but its not. I couldn't make any guess.
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    Graduate Measure theoretically equivalent

    When are two or more measures said to be measure theoretically equivalent? I have spent sometime on searching it; I am not getting it. Please someone help.
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    Graduate Is f(x)=x^a a Strictly Convex Function for a>1 on (0,\infty)?

    Is it true that f(x)=x^a is always a strictly convex function for a>1 on (0,\infty)?
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    Graduate Infimun and supremum of empty set

    Thanks Hurkyl and matt grime for your replies. Yes I got it now!
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    Graduate Infimun and supremum of empty set

    Why do we define(by convention) that infimum of an empty set as \infty and supremum as -\infty?
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    Graduate Solving L^p, L^q Subset Inequalities in X Sets of Arbitrary Size

    Thank you very much Billy Bob! I completely got it now. \int_{E_n}|f|^p\ge n\mu(E_n) Therefore, \mu(E_n)\le 1/n\int_{E_n}|f|^p\le 1/n\int |f|^p So \mu(E_n)=0 for some n or \mu(E_n)\to 0 The first one implies that f is bounded and the second one implies X contains sets of arbitrarily small...
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    Graduate Solving L^p, L^q Subset Inequalities in X Sets of Arbitrary Size

    yes absolutely..sorry. Then how..?
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    Graduate Solving L^p, L^q Subset Inequalities in X Sets of Arbitrary Size

    Yes absolutely. If E_n=\{x:|f(x)|^p>n\} then E_n=\{x:|f(x)|>n^{1/p}\} and E_1\subset E_2\subset...} Moreover, since X does not contain sets of arbitrarily small measure, \exists \epsilon >0 s.t. \mu(E)\ge \epsilon for all E\subset X From these facts I m unable to figure it out. Thanks...
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    Graduate Solving L^p, L^q Subset Inequalities in X Sets of Arbitrary Size

    Yes using the fact that p<q, I proved that \int |f|^q<\infty on {|f|\le 1 But I don't know how to make use of the fact the X doesn't contain sets of arbitrarily small positive measure in proving (2).
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    Graduate Solving L^p, L^q Subset Inequalities in X Sets of Arbitrary Size

    I don't know how is the above trure. We know only that there is a function f in L^p but not in L^q. From this we have to show that X contains sets of arbitrarily small measure.
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    Graduate Solving L^p, L^q Subset Inequalities in X Sets of Arbitrary Size

    Suppose 0 < p < q < \infty. Then L^p \nsubseteq L^q iff X contains sets of arbitrarily small positive measure. I have proved one part, namely, if X contains sets of arbitrarily small positive measure then L^p \nsubseteq L^q Can anyone give some hints to solve the other part? Thanks
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    Max diff entropy(Info. theory)

    If X is continuous r.v. and has pdf only in the positive real axis with E[X]=\alpha, E[X^2]=\beta, what is max diff. entropy of X? Thanks