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Consider the following random variable:

  • $X = 1$ if the flip of the coin is a head
  • $X = 0$ if the flip of the coin is a tail

How the above definition of random variable is different from

  • $X = 1$ if the flip of the coin is a head
  • $X = 1.01$ if the flip of the coin is a tail

What would be a judicious choice of random variable in a coin toss experiment? How the values assigned to the outcomes matters?

Can you please help me to understand the concept of random variable?

Vinod
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    I guess it depends on what you would want to calculate with this. Do you want to, for example, count how many heads/tails you get in a repeated experiment? – Matti P. Oct 19 '23 at 09:30
  • I don't understand. "judicious" with respect to what goal? Indicator variables, the first one you gave is an example, are often useful, but that doesn't mean you can't make other chocies. – lulu Oct 19 '23 at 09:30
  • @lulu Can you give an example of a goal in a coin toss experiment. In which situation $X=0$ and $X=1$ helpful. – Vinod Oct 19 '23 at 09:33
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    Sure, counting. If you flip the coin a thousand times, add up the observed values of $X$, and get $467$, then that means you got $467$ Heads. But, of course, you could add up the second one as well and then get the number of Heads via algebra. – lulu Oct 19 '23 at 09:34
  • But, it's your variable. You can assign values to it however you like. – lulu Oct 19 '23 at 09:35
  • Note that in more complex settings, you can't always untangle the total. For instance, if you are tossing a fair die. The indicator variable for the $6$ still counts the $6's$ but if your variable gave $1$ for a $6$, $.5$ for a $5$, and $0$ for the others, then if you tossed it $10$ times and got a score of $3$, you wouldn't know how many $6's$ you had. But, of course, that might not have been your goal. Maybe your goal was to determine how "happy" the outcome made you, and a $6$ makes you twice as happy as a $5$ and the other scores don't make you happy at all. – lulu Oct 19 '23 at 09:37
  • @lulu I doubt I understood random variables in a correct way. Can you point out references on how to design random variables related to a problem at hand. – Vinod Oct 19 '23 at 09:38
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    As a general rule, the problem you are looking at clarifies the variables. If I am interested in how many Heads I threw, it's natural to assign a variable that counts them. If I am looking at something more graded than a count, my variable should reflect that. But, in all cases, the problem comes first. – lulu Oct 19 '23 at 09:40
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    That is to say, variables measure things. If you are interested in the distribution of the heights of trees in a forest, I should have a variable that records the height of an observed tree. If I am interested in counting, I should have a variable that records the presence of the thing I want to count. Again, the variable should be linked to its purpose...to whatever it is you are interested in studying. – lulu Oct 19 '23 at 09:42
  • Your first choice of $X$ seems more sensible for coin tosses simply because $E(X) = .5$ if the coin is fair. – AlgTop1854 Oct 21 '23 at 01:45

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Remember the definition of a random variable. It is a function on the sample space. That means each function is a new random variable.(being a finite set I am ignoring measurability issues).

If your sample space is students in a college and you are taking one person at random, then you can measure their weight in kg which could be a number anywhere between 40 to 100 or something like that with some mean and some variance. (Think of the function that assigns weight to every individual). Or it could be age which is a different random variable with a much narrower band perhaps taking values mostly between 17 and 25. For every outcome (an element of sample space) what you are interested and what you choose to track is one random variable. One simplistic view of random variable is a numerical measurement on each outcome of a random experiment.

P Vanchinathan
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