Linked Questions
15 questions linked to/from "Absolutely continuous random variable" vs. "Continuous random variable"?
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Absolute continuity of distributions --- why do we need this? [duplicate]
Why do we define continuous probability distribution as those with absolutely continuous CDF, instead of just continuous CDF?
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2
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635
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How to prove some statistical model to be dominated? [duplicate]
I have a model following the PDF $$f_\theta(x) = \theta^2 x \exp(-\theta x)\delta_{[0, +\infty)}(x)$$ where $\theta > 0$ is the parameter and $\delta_\mathcal{S}$ denotes the indicator function of ...
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Sufficient Statistic for Absolutely Continuous Distribution [duplicate]
The following is a homework problem. Please tell me if my solution is correct and if not please point out my mistakes.
Let $x_{1}, x_{2},...,x_{M}$ be i.i.d. samples from the absolute continuous ...
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1
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0
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Are there really many continous distributions without the PDF [duplicate]
This is one interesting question I take some time to search if there is any distribution function that is continuous but without the PDF.
After some search I found Cantor distribution, sometimes ...
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Confusion about several different definitions of continuous random variables [duplicate]
It seems like different books define continuous random variables differently.
pdf definition:
The random variable X is continuous if a nonnegative function f exists, that is defined for all $x \in (-...
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10
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5
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Discrete and Continuous variables. What is the definition?
The definition of a continuous variable in our class seems to be, well, not a definition, as there are exceptions not included in its definition.
I am a 4th year math student and find it appalling ...
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23
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1
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How to sample from Cantor distribution?
What would be the best way to sample from Cantor distribution? It only has cdf and we can't invert it.
Tim♦
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12
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2
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If all the marginal distributions are continuous, then the joint distribution is continuous?
Consider a random vector $X\equiv (X_1,...,X_L)$. Assume that each $X_l$ is continuously distributed with support $\mathbb{R}$, for $l=1,...,L$. Does this imply that also $X$ should be continuously ...
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What is the definition of probability on a continuous distribution?
If I have a continuous distribution, $N$.
I say $f$ is a sample from $N$, $f \sim N$.
Now I want to determine the probability of $f$ having a value of $x$:
$$
P(x)=\lim_{t\rightarrow0} {\frac{\text{...
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2
votes
1
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514
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What is the difference between a probability measure and a probability density function? [duplicate]
During my research, I have repeatedly come across the terms probability measure and probability density function (pdf). I am familiar with the concept of a pdf, but I am not entirely sure how ...
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3
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Prove that a distribution is continuous at a point if and only if it has zero probability at that point
Let $X$ be a random variable with distribution function $F$.
Prove that $F$ is continuous at $x = a$ if and only if $\mathbb{P}(X = a) = 0$.
Could anyone give some hints? I am wondering where I ...
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Continuous distribution when there are flat regions
Consider a distribution function $F:\mathbb{R}\rightarrow [0,1]$ definining the positive, finite measures $\mu_F$ determined by
$$
\mu_F((a,b])\equiv F(b)-F(a)
$$
for each $a,b\in \mathbb{R}$ with $b&...
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0
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0
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132
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Singular Distributions in Statistics
Let $X$ be a continuous random variable which induces a probability measure on $\mathbb{R}$ denoted by $\mu$. Are there any instances in statistics when we deal with random variables $X$ such that $\...
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In real-life applications, which continuous distributions have NON-CONVERGENT expectations that require Lebesgue integration?
When computing expected values, Riemann integration works for only random variables with bounded support sets. For distributions with unbounded support sets, we can use improper Riemann integrals for &...
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Implications of continuity of a distribution function
Consider a random vector $(X,Y)$ with distribution function $F:\mathbb{R}^2\rightarrow [0,1]$ and let $\mu_F$ be the associated measure.
Take any $(a,b)\in \mathbb{R}^2$ and consider the set
$$
\...
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