[THOUGHT] We need to find the constant a using the total probability = 1, then compute Var((1/a)X) = (1/a^2) Var(X). We'll compute pmf values, check they sum to 1, solve for a, then compute E[X], E[X^2], Var(X), and finally Var((1/a)X) with Python. [/THOUGHT] [PYTHON] from fractions import Fraction

Compute probabilities for x = 0,1,2,3

probs = {x: abs(2*x - 1)/12 for x in range(4)} print("Probs x=0..3:", probs, "Sum:", sum(probs.values()))

Solve for a from total probability = 1

sum_0_3 = sum(probs.values()) a = 1 - sum_0_3 print("a =", a)

Double-check a is nonzero

assert a != 0

Define full pmf including x=4

probs[4] = a print("Full pmf:", probs, "Total sum:", sum(probs.values()))

Compute E[X] and E[X^2]

EX = sum(x * p for x, p in probs.items()) EX2 = sum((x2) * p for x, p in probs.items()) VarX = EX2 - EX2 print("E[X] =", EX) print("E[X^2] =", EX2) print("Var(X) =", VarX)

Compute Var((1/a)X) = (1/a^2) Var(X)

Var_scaled = VarX / (a**2) print("Var((1/a) X) =", Var_scaled) [/PYTHON]

[THOUGHT] Using the computed value a = 1/6, we have Var(X) = 5/4, so Var((1/a)X) = Var(6X) = 36 * 5/4 = 45. \boxed{45}