I'm taking an elementary probability course and came across this problem. I know what a binomial distribution is and its properties. But I have no idea how to prove if a given distribution is one.
Given distribution:
Just saying that the given distribution has discrete values and we only have 2 states ("success" and "fail") is enough?
PS: I'm not a native speaker, so if anything was cloudy, I can clarify them.
You can prove that using the moment generating function (MGF). Note that if $X\sim$ Binomial$(n,p)$, the MGF of $X$ is $(pe^t+1-p)^n$. In your problem, based on the probability distribution, the MGF of the random variable is $0.064+0.288e^t+0.432e^{2t}+0.216e^{3t}=(0.6e^t+0.4)^3$, so indeed, this is a binomial distribution with $n=3$ and $p=0.6$.