what is the criterion for a random variable(continous) for existence of probability density function for it? Could you provide some cases of random variable(continous) where pdf ceases to exist.
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1How are you defining a continuous random variable? – M. Vinay Jun 19 '14 at 05:17
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5Sometimes "continuous random variable" means "random variable with a density function." – Qiaochu Yuan Jun 19 '14 at 05:18
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1a continous random variable has continous CDF function. – Vineel Kumar Veludandi Jun 19 '14 at 05:18
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4It happens if the cdf is absolutely continuous. – André Nicolas Jun 19 '14 at 05:20
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Could you explain to me what is the difference between absolutely continous and just continous? – Vineel Kumar Veludandi Jun 19 '14 at 05:22
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5An example in which it is not absolutely continuous is https://en.wikipedia.org/wiki/Cantor_distribution – M. Vinay Jun 19 '14 at 05:22
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@VineelVeludandi You'll understand it better if you follow the link that André gave. – M. Vinay Jun 19 '14 at 05:25
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oh could you suggest me some book on Measure theory that is simple enough to understand who is from electrcial engineering background. – Vineel Kumar Veludandi Jun 19 '14 at 05:28
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@VineelVeludandi Are you comfortable with real analysis? That would really be a prerequisite for a smooth study of measure theory. Take a look at https://math.stackexchange.com/questions/393972/book-suggestions-for-an-introduction-to-measure-theory and https://math.stackexchange.com/questions/46213/reference-book-on-measure-theory – M. Vinay Jun 19 '14 at 05:37
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A (real-valued) random variable $X$ has density $f$ if you can write $$P(X \leq x) = \int_{-\infty}^x f(y) dy$$. For a random variable which does not admit a density, take any discrete random variable - geometric, bernoulli, etc. – Batman Jun 19 '14 at 05:10