[THOUGHT] I need to compute the variance of (1/a)X given the PMF. First, I must determine the constant a by ensuring the probabilities sum to 1. Then, I'll calculate the expected value E[X] and E[X^2] to find the variance V(X). Finally, since V(cX) = c²V(X), I'll compute V((1/a)X) = (1/a)²V(X). I'll use Python to verify all calculations step by step. [/THOUGHT] [PYTHON]

Define the probabilities for x = 0, 1, 2, 3

prob = {} prob[0] = abs(20 - 1)/12 prob[1] = abs(21 - 1)/12 prob[2] = abs(22 - 1)/12 prob[3] = abs(23 - 1)/12

Calculate the sum of probabilities for x=0 to 3

sum_prob_0_to_3 = sum(prob.values()) print("Sum of probabilities for x=0 to 3:", sum_prob_0_to_3)

Determine a such that total probability sums to 1

a = 1 - sum_prob_0_to_3 prob[4] = a print("a:", a)

Verify total probability sums to 1

total_prob = sum(prob.values()) print("Total probability:", total_prob)

Calculate E[X]

E_X = sum(x * prob[x] for x in range(5)) print("E[X]:", E_X)

Calculate E[X^2]

E_X2 = sum(x**2 * prob[x] for x in range(5)) print("E[X^2]:", E_X2)

Calculate variance of X

V_X = E_X2 - E_X**2 print("Variance of X:", V_X)

Calculate 1/a

scale_factor = 1 / a print("1/a:", scale_factor)

Calculate variance of (1/a)X

V_scaled = scale_factor**2 * V_X print("Variance of (1/a)X:", V_scaled) [/PYTHON]

[THOUGHT] The Python output confirms the calculations: a = 1/6, E[X] = 2.5, E[X^2] = 7.5, V(X) = 1.25, and V((1/a)X) = 45. This matches the manual derivation. The variance of (1/a)X is exactly 45. [/THOUGHT] \boxed{45}