Q: Discrete Probability True/False Classify as true or false and explain why for each discrete probability scenario
(a) If $X,Y$ are discrete random variables (both with a finite range) satisfying :
$∀a∈R$ $Pr(X = a) ≥ Pr(Y = a)$
,then it is always true that $E(X) = E(Y)$.
(discrete random variables are those whose probability distribution function is discrete - for example the
number of heads in n tosses of a fair coin is a discrete random variable)
(b) If $X,Y$ are two discrete random variables (both with a finite range) satisfying $E(X) > E(Y)$, then it is
always true that :
$∃a ∈ R Pr(X = a) < Pr(Y = a) $
(c) For a discrete random variable $X$ with a finite range, knowing $E(X)$ suffices to figure out the probability
distribution of X.
For part a and b, I'm lost on how to derive the answer, as I'm having trouble visualizing and comparing the random variables, considering they are describing two random events (The closest I've come is trying to draw up a dice rolling scenario, where $E(x) and E(y)$ are the expetced outcomes of the dice).
For part c, I'm leaning towards true, as knowing the expected value of a random variable can allow you to calculate the probability distribution
Any help would be greatly appreciated

A: a) and b) are both true and c) is false.
a): sum over $a$ you get $1=\sum P(X=a) \geq \sum P(Y=a)=1$. This implies that you must have $P(X=a)=P(Y=a)$ for all $a$ so $X$ and $Y$ have the same distribution, hence the the same mean.
b) follows from a). (Prove by contradiction).
I will leave c) to you. There are very simple counter-examples.
[Hint: Think of two different symmetric distributions].
