From other materials that I've read, the probability density of a continuous random variable must itself be continuous. Is this correct? If it is, I don't understand why that would be so, why can't the probability change abruptly?
Must probability density be continuous?
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2If I recall correctly, there is a restriction to at most countable discontinuities. – 2012-05-26
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1At least one discontinuity is common in densities of *practical* importance. – 2012-05-26
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6@DrewChristianson: Perhaps you are thinking of probability *distribution* functions. A density function can even be everywhere discontinuous. – 2012-05-26
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1"Continuous" distribution means the cdf (cumulative distribution function) is continuous. This does not mean the density is continuous, or even that a density exists. – 2012-05-27
4 Answers
Take $f(x) = 2x$, $0\le x \le 1$, and 0 otherwise. This is a density function which is not continuous.
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0I thought so. Thanks! – 2012-05-26
Michael Chernick asks for an example of a probability distribution with a density that is everywhere discontinuous.
As discussed in this question, there exists a measurable set $A \subset \mathbb{R}$ such that for every interval $I$, we have $0 < m(A \cap I) < m(I)$, and moreover $m(A) < \infty$. Then $f(x) = \frac{1}{m(A)} 1_A(x)$ is a nonnegative measurable function with $\int_\mathbb{R} f(x)\,dx = 1$, so it can be taken as the density of a continuous probability distribution. $f$ is nowhere continuous because every interval contains points of $A$ and $A^C$. Moreover, any function $g$ with $f=g$ a.e. is also nowhere continuous.
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3(+1) More generally, a construction similar to the following should work: Let $f$ be a probability density function continuous on $\mathbb R$. Take $g = f \cdot 1_{(\mathbb R \setminus \mathbb Q)}$. – 2012-05-27
Although valid I don't think the triangular density given by ncmathsadist is a good example. The U[0.1] density is discontinuous too because of the abrupt rise at x=0 and drop at x=1.But both these densities are continuous within their domain. I think a better example would be one with a discontinuity in its domain. Consider the density f(x)=2x for 0<=x<=1/2
and f(x)=(5-2x)/16 for 1/2
Certainly a density can have many such discontinuities. But can anyone give an example of a true probability distribution with a density that is everywhere discontinuous?
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2In some sense, *all* probability distributions that are absolutely continuous have (a *version* of) a density that is everywhere discontinuous on the closure of its support. – 2012-05-27
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0@cardinal: Could you please supply a reference or post an answer to explicate your assertion? Thank you. – 2015-12-14
No, need not be. However, the cumulative density function (CDF), is always continuous (mayn't be differentiable though) for a continuous random variable. For discrete random variables, CDF is discontinuous.
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0First, CDF is not cumulative density function. CDF is cumulative distribution function. Second, question is asking about probability density functions (PDF's) not about CDF's. – 2017-02-15