Independence of the conditions

In summary, the conversation is about a paper discussing a variable q that must characterize a stochastic process, have a domain of real numbers, and be stochastic. The question arises if "random variable" and "stochastic variable" are synonymous and if the three conditions are independent of each other. It is also unclear if q is a random variable or a stochastic process. The context and intended audience of the paper may influence the interpretation of the terms used.
  • #1
nomadreid
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Working through a paper about whose rigor I have my doubts, but I am always glad to be corrected. In the paper I find the following:
"We now investigate the random variable q. There are the following restrictions on q:
1) The variable q must characterize a stochastic process in the test interval
] ti - τ; ti + τ [ as τ tends to zero*;
2) the domain of q is the set of real numbers
3) q must be stochastic"
Two questions: First, "random variable" and "stochastic variable" are synonymous, no? So either (3) or the beginning assumption that q is a random variable would appear to be superfluous.
Second, don't (3) and (2) together imply (1)?
In any case, somehow it seems that these three conditions are not completely independent of one another. I would be grateful for any indications to confirm or deny my intuition.

*PS. The original actually said
"1) The variable q must characterize a stochastic process in the test interval
] ti - τ0; ti + τ [ as τ tends to zero"; but I think that the τ0 is a typo.
 
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  • #2
The wording is confusing. Is q a random variable or a stochastic process (a function with random variable range)?
 
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  • #3
nomadreid said:
Second, don't (3) and (2) together imply (1)?

I'd say no, but the phrase "characterize a stochastic process" isn't defined.

A stochastic process produces random trajectories. As a literal example, tossing a paper air plane across a drafty room might be modeled a stochastic process. Let v(t) be a specific trajectory that is realized by a stochastic process. Then there is nothing stochastic about v(t). It is a definite trajectory. It's properties, such as ##lim_{t\rightarrow 2} v(t)## are not stochastic. However, we can define a random variable ##Q## by saying: A realization of ##Q## consists of taking a random trajectory v(t) generated by the stochastic process and finding the value of ##q = lim_{t\rightarrow 2} v(t) ##. This makes ##Q## a random variable by virtue of the fact that ##v(t)## is chosen at random.

If we only say that ##Q## is a random variable and that the domain of ##Q## is the set of real number then this does not imply the structure that ##Q## was defined a function of a random trajectories.
 
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  • #4
Thank you, mathman and Stephen Tashi.
mathman said:
The wording is confusing. Is q a random variable or a stochastic process (a function with random variable range)?
Stephen Tashi said:
I'd say no, but the phrase "characterize a stochastic process" isn't defined.
Yes, his wording is confusing, and of course a problem is that I have quoted out of context. Apparently what he means is that q(t) =x(t) (x-position, t-time) is a stochastic process, and for a specific value of t = ti, q(ti)=q is a random variable, then integrating and differentiating over q.
Stephen Tashi said:
A stochastic process produces random trajectories. As a literal example, tossing a paper air plane across a drafty room might be modeled a stochastic process. Let v(t) be a specific trajectory that is realized by a stochastic process. Then there is nothing stochastic about v(t). It is a definite trajectory. It's properties, such as limt→2v(t)lim_{t\rightarrow 2} v(t) are not stochastic. However, we can define a random variable QQ by saying: A realization of QQ consists of taking a random trajectory v(t) generated by the stochastic process and finding the value of q=limt→2v(t)q = lim_{t\rightarrow 2} v(t) . This makes QQ a random variable by virtue of the fact that v(t)v(t) is chosen at random.

If we only say that QQ is a random variable and that the domain of QQ is the set of real number then this does not imply the structure that QQ was defined a function of a random trajectories.
Thank you, Stephen Tashi, your explanation and counter-example make it clearer. One question about it though: in your example, v(t) is not a random variable, only QQ; in the conditions that I stated, however, he starts out saying that q is a random variable, and then in (3) saying that it must be stochastic: isn't this redundant? Would it have sufficed to either
(a) eliminate (3), or
(b) keep (3) but not start by saying that q was a random variable?
 
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  • #5
nomadreid said:
in the conditions that I stated, however, he starts out saying that q is a random variable,
In what you quoted, he starts out by saying that q must "characterize a stochastic process". It isn't clear whether that phrase implies q is a random variable.

To interpret what you quoted, it would be necessary to understand the context - which includes the intended audience for the paper. For example, articles about applied science sometimes use the terms "random" or "stochastic" in the common language sense of those words instead of in the mathematical sense of "random variable" or "stochastic process".
 
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  • #6
Thanks, Stephen Tashi. The intended audience is theoretical physicists; the author proposes an alternative derivation of Schrödinger's equation for a specific case of a classical (pre-quantum) particle in chaotic motion. He uses the term which is Russian for, literally, "stochastic variable", he investigates whether it is correlated with another "random variable", so I interpreted his usage to mean "random variable" in the mathematical sense.
 

What is the concept of "Independence of the conditions"?

Independence of the conditions refers to the idea that the outcome of an event or experiment is not affected by any external factors or conditions. In other words, the outcome is solely determined by the variables that are being studied and not influenced by any other factors.

Why is "Independence of the conditions" important in scientific research?

"Independence of the conditions" is important in scientific research because it allows for more accurate and reliable results. By controlling for external factors, scientists can isolate the variables they are studying and draw more accurate conclusions about cause and effect relationships.

How do scientists ensure "Independence of the conditions" in their experiments?

Scientists ensure "Independence of the conditions" by carefully designing their experiments and controlling for any potential external factors that could influence the results. This may involve using control groups, randomization, and other methods to eliminate any potential biases.

What are some examples of experiments that require "Independence of the conditions"?

Experiments that require "Independence of the conditions" include drug trials, psychological experiments, and studies in physics and chemistry. These experiments rely on controlling for external factors to accurately measure the effects of specific variables.

What are the potential consequences of not ensuring "Independence of the conditions" in an experiment?

If "Independence of the conditions" is not ensured in an experiment, the results may be inaccurate or misleading. This can lead to incorrect conclusions and potentially harmful consequences if the results are used to make important decisions or inform public policy.

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