In the simplest setting, a continuous random variable is one that can assume any value on some interval of the real numbers.
For example, a uniform random variable is often defined on the unit interval [0,1], which means that this random variable could assume any value between 0 and 1, including 0 and 1. Some possibilities would be 1/3, 0.3214, pi/4, e/5, and so on ... in other words, any of the numbers in that interval.
As another example, a normal random variable can assume any value between -infinity and +infinity (another interval). Most of these values would be extremely unlikely to occur but they would be possible. The random variable could assume values of 3, -10000, pi, 1000*pi, e*e, ... any possible value in the real numbers.
It is also possible to define continue random variables that assume values on the entire (x,y) plane, or just on the circumference of a circle, or anywhere that you can imagine that is essentially equivalent (in some sense) to pieces of a real line.
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It is a discrete random variable.
continuous random variable
Zero.
You integrate the probability distribution function to get the cumulative distribution function (cdf). Then find the value of the random variable for which cdf = 0.5.
A discrete random variable is a variable that can only take some selected values. The values that it can take may be infinite in number (eg the counting numbers), but unlike a continuous random variable, it cannot take any value in between valid results.