Try to find a discrete random variable that can take infinitely many values. For example, consider a situation when you flip a coin until you get tails. The amount of heads in a row by can be represented by a random variable X.
The statement is false. See solution.
Practice makes perfect
We will try to explain why the given statement is false.
Random variables that can take an infinite amount of values are continuous.
Let's say that you are bored and you start flipping a coin. You start counting how many times you can get tails in a row. If the coin lands on heads you start all over again.
This is a random event that can be represented by a random variable X.
X - Amount of Tails In a Row
Let's count the probability of getting 6056 tails in a row. Since the probability of flipping tails is 12 and the flips are independent, the probability of getting 6056 tails in a row is equal to ( 12)^(6056).
P(X=6056)=(1/2)^(6056)
The probability of this event happening is an incredibly small number, but it is still not zero and it technically could happen. Of course, instead of 6056 we could have chosen any different integer — we had infinite amount of possibilities!
X can take any integer
Therefore, X is a random variable which can take an infinite amount of values. However, it does not make sense to flip, for example, 34 tails in a row. Thus, X is not continuous.
X can take any integer and
X is not continuous
We found a counterexample to the statement given in the exercise, therefore it is a false statement! This is not the only counterexample — it was an example of a discrete random variable with a geometric distribution. There are many other (infinitely many) random variables that are discrete and also infinite.
