Why are random variables needed?

In summary: However, if you use a random variable, the mean and variance may not be exactly the same as if you had used the original distribution.
  • #1
dionysian
53
1
please corret me if i am incorrect in my understanding of a RV,PMF or anything else but as i understand it a random variable simply maps a expirmental outcome to a real number. And a probability mass function simply gives the probabilty that a number will occur.

Now my question is this: why can't we simply map a expiremental outcome to a probability rather than use a random variable to map from the sample space a to a real number then the map a real number to a probability via the PMF?
 
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  • #2
dionysian said:
please corret me if i am incorrect in my understanding of a RV,PMF or anything else but as i understand it a random variable simply maps a expirmental outcome to a real number. And a probability mass function simply gives the probabilty that a number will occur.

Now my question is this: why can't we simply map a expiremental outcome to a probability rather than use a random variable to map from the sample space a to a real number then the map a real number to a probability via the PMF?

a random variable is a function of the experimental outcomes. It is convenient not to redefine your probability distribution in terms of every random variable but to keep a fixed probability distribution and then look at the statistics of different random variables.
 
  • #3
Ahh this is what i suspected... So its fair to say that the RV helps keeps the pdf independent of each expirment
 
  • #4
dionysian said:
Ahh this is what i suspected... So its fair to say that the RV helps keeps the pdf independent of each expirment

yes.

Typically in statistics you look at sampling distributions. The sample mean and variance are example of random variables on the sampling distribution. If one knows something about the original distribution - say that is is normally distributed - then you know something about the distribution of the mean and the variance.
 

1. Why do scientists use random variables in their experiments?

Random variables are used in scientific experiments to introduce an element of chance or uncertainty. This allows researchers to study the effects of different variables on a system and to draw conclusions about cause and effect. Random variables also help to ensure the validity and reliability of experimental results.

2. How do random variables help in statistical analysis?

Random variables are essential in statistical analysis as they provide the basis for probability theory. By assigning numerical values to uncertain outcomes, random variables allow for the calculation of probabilities and the prediction of future events. They also make it possible to analyze and compare data sets, and to draw conclusions about populations based on samples.

3. What is the difference between discrete and continuous random variables?

Discrete random variables can only take on distinct, separate values, such as whole numbers, while continuous random variables can take on any value within a given range. This difference is important in statistical analysis, as different methods and formulas are used to analyze each type of variable.

4. Why are random variables important in modeling complex systems?

In complex systems, there are often many unknown variables and factors that can affect the outcome. Random variables allow for the introduction of random or uncertain factors into models, making them more realistic and accurate. They also help to identify the most influential variables and to predict the behavior of the system under different conditions.

5. Can random variables be controlled in experiments?

While the values of random variables cannot be controlled, they can be manipulated or selected to some extent. For example, researchers can control the sample size or the range of values for a continuous variable. This allows for the testing of hypotheses and the identification of relationships between variables, even though the values themselves are random.

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