Concept guide
Finite discrete probability distribution: validating a PMF, event probability, expectation and pushforward
A random variable X takes the values −1, 0, 1, 2 with masses p(−1)=1/6, p(0)=1/3, p(1)=1/4, p(2)=1/4. (1) First check that p is a valid PMF; (2) find P(X≥1); (3) find E[X]; (4) find the pushforward distribution of Y=X², prove it is again a PMF, find E[Y] and cross-check it against Σx²p(x); (5) decide that two candidates q and r are not PMFs: every mass of q lies between 0 and 1 but the total is not 1, while r sums to 1 but has a negative mass.