Intereting Posts

the knot surgery – from a $6^3_2$ knot to a $3_1$ trefoil knot
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Generalisation of Dominated Convergence Theorem
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Let $\rho(x,y)= \frac{d(x,y)}{1+d(x,y)}$, show that $\rho$ is a metric equivalent to $d$
Every subsequence of $x_n$ has a further subsequence which converges to $x$.Then the sequence $x_n$ converges to $x$.
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What exactly is the difference between weak and strong induction?
Reference for the subgroup structure of $PSL(2,q)$
Finding derivative of $\sqrt{x}$ using only limits
Computing $\int (1 – \frac{3}{x^4})\exp(-\frac{x^{2}}{2}) dx$
Set of open intervals in R with rational endpoints is a basis for standard topology on R
Showing that if $R$ is a commutative ring and $M$ an $R$-module, then $M \otimes_R (R/\mathfrak m) \cong M / \mathfrak m M$.
Parity of number of factors up to a bound?
General solution to wave equation of half-line with nonhomogeneous Neumann boundary

It seems that one usually use the set of all possible values of $X$ as the sample space of random variable $X$. (Therefore discrete random variables have countable sample space, continuous random variables have uncountable sample space.) However, I don’t think this is right. Since random variables are defined as measurable functions over sample space, the above assumption would make sum/product of a discrete random variable and a continuous random variable meaningless (because they are function over different spaces.)

Well, this post says that

Every uncountable standard Borel space is isomorphic to [0,1] with the Borel σ-algebra. Moreover, every non-atomic probability measure on a standard Borel space is equivalent to Lebesgue-measure on [0,1].

- Toss a fair die until the cumulative sum is a perfect square-Expected Value
- Laplace transform of integrated geometric Brownian motion
- Asymptotic behaviour of a two-dimensional recurrence relation
- Looking for intuition behind coin-flipping pattern expectation
- Calculate the expectation of a random variable given its cdf
- What is the joint distribution of two random variables?

However, this doesn’t solve all the problems.

First, that claim is true only for uncountable sample space, so the problem of adding of a discrete RV and a continuous RV is still unsolved.

Second, even with two continuous RV, there are still problems if we consider expectation (integration). Although the quoted claim says that *“every non-atomic probability measure on a standard Borel space is equivalent to Lebesgue-measure on [0,1].”*, however, two different measures cannot be transformed to Lebesgue measure on [0,1] with same isomorphism. Therefore two RVs $X$ and $Y$ may have different measures on [0,1], which will make $\mathbb{E}XY$ meaningless.

This really confused me…. Can people really base all different random variables onto one same sample space with one same measure?

- Combinatorics Distribution - Number of integer solutions Concept Explanation
- Find characteristic function of ZX+(1-Z)Y with X uniform, Y Poisson and Z Bernoulli
- Choose a random number between 0 and 1 and record its value. Keep doing it until the sum of the numbers exceeds 1. How many tries do we have to do?
- covariance of increasing functions
- Simplifying Multiple Integral for Compound Probability Density Function
- Variance of sum of random variables 2
- Expected Value of a Continuous Random Variable
- Hard Integral $\frac{1}{(1+x^2+y^2+z^2)^2}$
- Finding probability using moment-generating functions
- Prove a thm on stopped processes given fundamental principle 'you can't beat the system'?

It seems that one usually use the set of all possible values of X as the sample space of random variable X.

Well, your post is an excellent example of why one should not. Actually, the reason why some introductory probability courses in some countries focus so much on the sample space and (even worse) why they recommend to take as sample space the image set of $X$ is a real mystery since:

- this option is ludicrous in itself,
- it leads to confusion for anybody actually thinking about the problem,
- it is completely abandoned (rightly so) in later, higher, courses (not to mention that probabilists themselves refute it).

Terence Tao’s citation in Qiaochu Yuan’s answer (and t.b.’s very short comment on his own answer) on the page you link to, all say it well so I will be brief. The take-home message is that, in most situations, *there exists* some probability space $(\Omega,\mathcal F,P)$ such that the family of random variables one is interested in can be defined simultaneously, as long as their joint distribution are compatible. Kolmogorov’s consistency theorem gives you that, in a much wider setting than you would actually care for, and this is it. The nature of $(\Omega,\mathcal F,P)$ is left unspecified (although one knows that $[0,1]$ with its Borel sigma-algebra would fit an awful lot of situations) and, more importantly, it is irrelevant. All that counts is that *some* such probability space $(\Omega,\mathcal F,P)$ exists. Once one knows it does, one can turn to the actual probability questions one wants to solve.

Have a look at this post of Terry Tau:

http://terrytao.wordpress.com/2010/01/01/254a-notes-0-a-review-of-probability-theory/

And specifically:

” we will try to adhere to the following important dogma: probability theory is only “allowed” to study concepts and perform operations which are preserved with respect to extension of the underlying sample space. “

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- If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$
- Showing that rationals have Lebesgue measure zero.