Problem Analysis #36

Solar-Pro-2 (31B)(high) · Run 1

34.17s
6,659 toks
Problem Statement

A discrete random variable XX takes integer values from 00 to 44, and its probability mass function is given by

P(X=x)={∣2x−1∣12(x=0,1,2,3),a(x=4),\text{P}(X = x) = \begin{cases}\dfrac{|2x - 1|}{12} & (x = 0, 1, 2, 3), \\[4pt]a & (x = 4),\end{cases}

where aa is a nonzero constant. Compute

V(1aX).\text{V}\left(\dfrac{1}{a}X\right).

[3 points]\

Ground Truth
45
Execution Trace
Reasoning Process

[THOUGHT] To compute the variance of 1aX\dfrac{1}{a}X, we first determine the constant aa by ensuring the probability mass function sums to 1. Then, we calculate the expected value E[X]E[X] and E[X2]E[X^2] to find Var(X)\text{Var}(X). Finally, using the property Var(cX)=c2Var(X)\text{Var}(cX) = c^2 \text{Var}(X), we compute the desired variance.

  1. Find aa:
    • Sum probabilities for x=0,1,2,3x = 0, 1, 2, 3:
∑x=03∣2x−1∣12=112+112+312+512=1012=56\sum_{x=0}^3 \frac{|2x - 1|}{12} = \frac{1}{12} + \frac{1}{12} + \frac{3}{12} + \frac{5}{12} = \frac{10}{12} = \frac{5}{6}
  • Solve 56+a=1\frac{5}{6} + a = 1 to get a=16a = \frac{1}{6}.
  1. Calculate E[X]E[X]:
E[X]=∑x=04x⋅P(X=x)=0⋅112+1⋅112+2⋅312+3⋅512+4⋅16=52E[X] = \sum_{x=0}^4 x \cdot P(X = x) = 0 \cdot \frac{1}{12} + 1 \cdot \frac{1}{12} + 2 \cdot \frac{3}{12} + 3 \cdot \frac{5}{12} + 4 \cdot \frac{1}{6} = \frac{5}{2}
  1. Calculate E[X2]E[X^2]:
E[X2]=∑x=04x2⋅P(X=x)=0⋅112+1⋅112+4⋅312+9⋅512+16⋅16=152E[X^2] = \sum_{x=0}^4 x^2 \cdot P(X = x) = 0 \cdot \frac{1}{12} + 1 \cdot \frac{1}{12} + 4 \cdot \frac{3}{12} + 9 \cdot \frac{5}{12} + 16 \cdot \frac{1}{6} = \frac{15}{2}
  1. Compute Var(X)\text{Var}(X):
Var(X)=E[X2]−(E[X])2=152−(52)2=54\text{Var}(X) = E[X^2] - (E[X])^2 = \frac{15}{2} - \left(\frac{5}{2}\right)^2 = \frac{5}{4}
  1. Compute Var(1aX)\text{Var}\left(\dfrac{1}{a}X\right):
Var(1aX)=(1a)2Var(X)=62⋅54=36⋅54=45\text{Var}\left(\dfrac{1}{a}X\right) = \left(\dfrac{1}{a}\right)^2 \text{Var}(X) = 6^2 \cdot \frac{5}{4} = 36 \cdot \frac{5}{4} = 45

[/THOUGHT] [PYTHON]

Calculate the constant a

prob_sum = sum(abs(2*x - 1)/12 for x in range(4)) a = 1 - prob_sum

Calculate E[X]

E_X = sum(x * (abs(2*x - 1)/12 if x < 4 else a) for x in range(5))

Calculate E[X^2]

E_X2 = sum(x**2 * (abs(2*x - 1)/12 if x < 4 else a) for x in range(5))

Variance of X

var_X = E_X2 - E_X**2

Variance of (1/a)X

var_scaled = (1/a)**2 * var_X

print(var_scaled) [/PYTHON]

Tool Output
[PYTHON OUTPUT] 44.99999999999999 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] The Python output is approximately 45, which matches our manual calculation. The variance of 1aX\dfrac{1}{a}X is 45\boxed{45}. [/THOUGHT]

Final Model Output
45

Comparative Analysis (P36) (Average across runs)

Avg Time Taken

Gemini-3-Pro-Preview
684.4
Solar-Open-100B
623.1
K-EXAONE-236B-A23B
454.7
EXAONE-4.0.1-32B (high)
193.2
Deepseek-V3.2
67.4
Llama-VARCO-8B-Instruct
62.2
Kanana-2-30B-Thinking
54.1
Kanana-2-30B-Thinking-2601
44.8
HCX-007(high)
36.9
Solar-Pro-2 (31B)(high)
34.2
A.X-4.0 (72B)
25.7
Claude-Opus-4.5
14.8
Grok-4.1-fast
9.0
K-EXAONE-236B-A23B
8.8
GPT-5.1 (high)
8.3
GPT-5.2 (high)
7.3

Avg Token Usage

K-EXAONE-236B-A23B
97,464
EXAONE-4.0.1-32B (high)
43,263
Kanana-2-30B-Thinking-2601
15,983
Kanana-2-30B-Thinking
10,007
Solar-Pro-2 (31B)(high)
6,659
Solar-Open-100B
5,881
Gemini-3-Pro-Preview
5,840
HCX-007(high)
4,585
Grok-4.1-fast
3,971
Claude-Opus-4.5
3,958
Deepseek-V3.2
3,770
K-EXAONE-236B-A23B
3,020
GPT-5.1 (high)
2,953
Llama-VARCO-8B-Instruct
2,843
GPT-5.2 (high)
2,716
A.X-4.0 (72B)
2,312