Although I've taken probability and statistics as a course (numerous times, either in college or via MOOCs), I never was quite certain about the definition of a random variable.
However, now I'm more cognizant thanks to the MIT course on Probability. So basically, a Random variable is a function that maps the outcome of a probabilistic experiment to a measure. An example will make this clear. If I have 30 students in a class and the experiment is simply to pull out a student at random, then the probability of picking one student is 1/30. However, each student has attributes: like height, weight, GPA, etc. If I then map that student to their height, that is the random variable. So based on the outcome of the experiment (i.e., who I choose), the mapping from the student to the height (or weight, or GPA) is a random variable!
So a random variable maps from the sample space (omega) to a number. You can combine random variables and that itself is a random variable. Every outcome will be mapped to a number and the mapping can be discrete or continuous.
Now, the pmf (probability mass function, a term that is only used when the random variable in discussion takes on discrete values) is a probability law or probability distribution of the random variable X. So it describes the probabilities given to the random variable when the random variable takes on a certain values. So P[X=1] = some probability; P[X=2] = another probability; P[X=3] = yet another probability. An exhaustive description like this constitutes the pmf.
However, now I'm more cognizant thanks to the MIT course on Probability. So basically, a Random variable is a function that maps the outcome of a probabilistic experiment to a measure. An example will make this clear. If I have 30 students in a class and the experiment is simply to pull out a student at random, then the probability of picking one student is 1/30. However, each student has attributes: like height, weight, GPA, etc. If I then map that student to their height, that is the random variable. So based on the outcome of the experiment (i.e., who I choose), the mapping from the student to the height (or weight, or GPA) is a random variable!
So a random variable maps from the sample space (omega) to a number. You can combine random variables and that itself is a random variable. Every outcome will be mapped to a number and the mapping can be discrete or continuous.
Now, the pmf (probability mass function, a term that is only used when the random variable in discussion takes on discrete values) is a probability law or probability distribution of the random variable X. So it describes the probabilities given to the random variable when the random variable takes on a certain values. So P[X=1] = some probability; P[X=2] = another probability; P[X=3] = yet another probability. An exhaustive description like this constitutes the pmf.
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