what is Pr[X = 2]?

Question

Let X be a random variable that takes on the values -1, 1, and 2 . If Pr[X = 1] = .7 and E[X] = .7, then what is Pr[X = 2]?

Answer 1

E[X] = .7 is the weighted average of the sum of each probability times the outcome

E[X] = .7 = Pr(X=1)(1) + Pr(X=-1)(-1) + Pr(X=2)(2 = .7 - Pr(x=-1) + 2Pr(x=2)

.7 = .7 - Pr(x=-1) + 2Pr(x=2)

Pr(x=-1) = 2Pr(x=2)

Pr(x=1) + Pr(x=-1) + Pr(x=2) = 1. The probabilities sum to one.

.7 = 1- Pr(x=-1) - Pr(x=2)

Pr(x=-1) + Pr(x=2) = 1-.7 = .3

Pr(x=-1) -2Pr(x=2) = 0

subtract those last 2 equations to get

3Pr(x=2) = .3

divide by 3

Pr(x=2) = .1

Pr(x=-1) = .2

Pr(x=1) = .7

those 3 probabilities sum to one

Pr(x=2) = .1