Why are random variables needed?

[dionysian]

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 probability 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?

 

[wofsy]

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 probability 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.

 

[dionysian]

Ahh this is what i suspected... So its fair to say that the RV helps keeps the pdf independent of each expirment

 

[wofsy]

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.