[THOUGHT] Let me solve this step by step.

First, I need to find the value of $a$ using the fact that the sum of all probabilities must equal 1.

For $x = 0, 1, 2, 3$:

• $P(X = 0) = \frac{|2(0) - 1|}{12} = \frac{1}{12}$
• $P(X = 1) = \frac{|2(1) - 1|}{12} = \frac{1}{12}$
• $P(X = 2) = \frac{|2(2) - 1|}{12} = \frac{3}{12}$
• $P(X = 3) = \frac{|2(3) - 1|}{12} = \frac{5}{12}$

And $P(X = 4) = a$.

The sum must equal 1: $\frac{1}{12} + \frac{1}{12} + \frac{3}{12} + \frac{5}{12} + a = 1$

Then I need to compute $V\left(\frac{1}{a}X\right) = \frac{1}{a^2}V(X)$.

Let me calculate this using Python. [/THOUGHT] [PYTHON] from fractions import Fraction

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

probs = {} for x in range(4): probs[x] = Fraction(abs(2*x - 1), 12) print(f"P(X = {x}) = {probs[x]}")

Sum of probabilities for x = 0, 1, 2, 3

sum_probs = sum(probs.values()) print(f"\nSum of P(X=0) to P(X=3) = {sum_probs}")

Find a

a = 1 - sum_probs print(f"a = P(X = 4) = {a}") probs[4] = a

Verify sum = 1

total = sum(probs.values()) print(f"Total probability = {total}")

Calculate E(X)

E_X = sum(x * probs[x] for x in range(5)) print(f"\nE(X) = {E_X}")

Calculate E(X^2)

E_X2 = sum(x**2 * probs[x] for x in range(5)) print(f"E(X^2) = {E_X2}")

Calculate V(X) = E(X^2) - (E(X))^2

V_X = E_X2 - E_X**2 print(f"V(X) = {V_X}")

Calculate V((1/a)X) = (1/a^2) * V(X)

V_result = V_X / (a**2) print(f"\nV((1/a)X) = V(X) / a^2 = {V_result}") print(f"V((1/a)X) as decimal = {float(V_result)}") [/PYTHON]

[THOUGHT] The calculation shows that:

• The probabilities for x = 0, 1, 2, 3 are 1/12, 1/12, 1/4, 5/12 respectively
• The constant a = 1/6
• E(X) = 5/2
• E(X²) = 15/2
• V(X) = 5/4
• V((1/a)X) = V(X)/a² = (5/4)/(1/36) = (5/4) × 36 = 45

The answer is 45. [/THOUGHT]

\boxed{45}