GRE exam question: Determine which of the following is a random variable.

Question

As I know it,

A Random Variable is a variable which represents the outcome of a random experiment.

Is this correct?
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According to the above definition, the following is not a Random Variable as, in this case, the count is deterministic.

1. We flip a coin thrice and a random variable $X$ represents the count of the flip. I.e. $X = 1, 2, 3$.

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The following is a Random Variable as the total count of Heads in tosses is indeterministic.

2. We flip a coin thrice and a random variable $X$ represents the total count of Heads in THREE flips. I.e. $X = 0, 1, 2, 3$.

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But, what about the following? Is the following an example of a random variable?

3. We flip a coin thrice and a random variable $X$ represents the probability of occurring Heads in each flip.

Answer 1

Not. A probability is a specific number ($\left(\frac12\right)^3$ in the case of a fair coin).

Answer 2

1. Although more often viewed as a constant, technically this is a random variable.   It has a degenerate distribution; meaning that, in this case, it equals 3 with certainty.   $\mathsf P(X{=}k)=\mathbf 1_{k=3}$.   ie: the probability equals $1$ if $k=3$ or equals $0$ otherwise.

2. A much clearer case; this is a random variable with a binomial distribution. $\mathsf P(X{=}k)= {^3\mathrm C_k} 2^{-3}\mathbf 1_{k\in\{0,1,2,3\}}$

3. This is an event.   It directly describes a set of outcomes of the experiment's sample space, without mapping them to a measure.   $\mathbb P(X)=2^{-3}$ since $X=\{({\sf H,H,H})\}$

Answer 3

I would say that the probability of seeing $n$ heads in the first $n$ coin flips is not a random variable because it is not something that we can measure or calculate from the results of the experiment.

The frequency of seeing $n$ heads in the first $n$ coin flips is a random variable because we can measure that. But the probability of seeing $n$ heads in the first $n$ coin flips can only be determined if we have an a priori model of the coin's behaviour.