Evaluation

Input	Output
What does the following represent: $\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi}$	Anthropic: Claude 3 Opus PublicPass B The given integral expression, $\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi}$ , is a famous definite integral known as the Gaussian integral or the Euler-Poisson integral. It represents the area under the curve of the function $e^{-x^2}$ over the entire real line, from negative infinity to positive infinity. Here are some key points about this integral: The function $e^{-x^2}$ is an even function, meaning that it is symmetric about the y-axis. This symmetry is crucial in evaluating the integral. The function $e^{-x^2}$ is a non-negative function, as the exponential function is always positive, and squaring x ensures that the exponent is always non-positive. The function $e^{-x^2}$ approaches 0 as x approaches either positive or negative infinity, ensuring that the improper integral converges. The Gaussian integral is a key component in probability theory and statistics, as it is closely related to the normal distribution (also known as the Gaussian distribution). The probability density function (PDF) of the standard normal distribution is given by: $f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ The constant factor $\frac{1}{\sqrt{2\pi}}$ ensures that the total area under the PDF curve is equal to 1, which is a requirement for any valid probability distribution. The Gaussian integral can be evaluated using various techniques, such as double integration with polar coordinates, or by using the Gamma function. In summary, the expression $\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi}$ represents the area under the curve of $e^{-x^2}$ over the entire real line, and it is a fundamental result in mathematics with significant applications in probability theory and statistics.

Input

Output

What does the following represent: $\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi}$

PublicPass

The given integral expression, $\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi}$ , is a famous definite integral known as the Gaussian integral or the Euler-Poisson integral. It represents the area under the curve of the function $e^{-x^2}$ over the entire real line, from negative infinity to positive infinity.

Here are some key points about this integral:

The function $e^{-x^2}$ is an even function, meaning that it is symmetric about the y-axis. This symmetry is crucial in evaluating the integral.
The function $e^{-x^2}$ is a non-negative function, as the exponential function is always positive, and squaring x ensures that the exponent is always non-positive.
The function $e^{-x^2}$ approaches 0 as x approaches either positive or negative infinity, ensuring that the improper integral converges.
The Gaussian integral is a key component in probability theory and statistics, as it is closely related to the normal distribution (also known as the Gaussian distribution). The probability density function (PDF) of the standard normal distribution is given by:

$f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$

The constant factor $\frac{1}{\sqrt{2\pi}}$ ensures that the total area under the PDF curve is equal to 1, which is a requirement for any valid probability distribution.
The Gaussian integral can be evaluated using various techniques, such as double integration with polar coordinates, or by using the Gamma function.

In summary, the expression $\int_{-\infty}^\infty e^{-x^2} \,dx = \sqrt{\pi}$ represents the area under the curve of $e^{-x^2}$ over the entire real line, and it is a fundamental result in mathematics with significant applications in probability theory and statistics.

Evaluation ID: f987ce86-114e-4319-b0cb-f6ef54cc7345
Created: April 23, 2024

Evaluation Rating

Binary Pass/Fail: Pass

Score: 100

Additional Feedback: The submitted answer is a superset of the expert answer and is fully consistent with it.

Factual Consistency: B: the submitted answer is a superset of the expert answer and is fully consistent with it

Raw Output

{
  "id": "gen-fSxPoy4teqbQniGNBOJ5Ln1TFHS1",
  "model": "anthropic/claude-3-opus",
  "object": "chat.completion",
  "created": 1713841616,
  "choices": [
    {
      "index": 0,
      "message": {
        "role": "assistant",
        "content": "The given integral expression, $\\int_{-\\infty}^\\infty e^{-x^2} \\,dx = \\sqrt{\\pi}$, is a famous definite integral known as the Gaussian integral or the Euler-Poisson integral. It represents the area under the curve of the function $e^{-x^2}$ over the entire real line, from negative infinity to positive infinity.\n\nHere are some key points about this integral:\n\n1. The function $e^{-x^2}$ is an even function, meaning that it is symmetric about the y-axis. This symmetry is crucial in evaluating the integral.\n\n2. The function $e^{-x^2}$ is a non-negative function, as the exponential function is always positive, and squaring x ensures that the exponent is always non-positive.\n\n3. The function $e^{-x^2}$ approaches 0 as x approaches either positive or negative infinity, ensuring that the improper integral converges.\n\n4. The Gaussian integral is a key component in probability theory and statistics, as it is closely related to the normal distribution (also known as the Gaussian distribution). The probability density function (PDF) of the standard normal distribution is given by:\n\n   $f(x) = \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}}$\n\n   The constant factor $\\frac{1}{\\sqrt{2\\pi}}$ ensures that the total area under the PDF curve is equal to 1, which is a requirement for any valid probability distribution.\n\n5. The Gaussian integral can be evaluated using various techniques, such as double integration with polar coordinates, or by using the Gamma function.\n\nIn summary, the expression $\\int_{-\\infty}^\\infty e^{-x^2} \\,dx = \\sqrt{\\pi}$ represents the area under the curve of $e^{-x^2}$ over the entire real line, and it is a fundamental result in mathematics with significant applications in probability theory and statistics."
      },
      "finish_reason": "end_turn"
    }
  ],
  "usage": {
    "prompt_tokens": 43,
    "completion_tokens": 467,
    "total_tokens": 510,
    "total_cost": 0.03567
  }
}

Raw Rating Output

{
  "id": "gen-sRHFTnuyUZ0Tj45HjjLnC8fRzXjc",
  "model": "openai/gpt-3.5-turbo-0125",
  "object": "chat.completion",
  "created": 1713841622,
  "choices": [
    {
      "index": 0,
      "message": {
        "role": "assistant",
        "content": null,
        "tool_calls": [
          {
            "id": "call_tLT32JfT5hQQMRescOz4qqtp",
            "type": "function",
            "function": {
              "name": "get_eval_rating",
              "arguments": "{\"binary_pass_fail\":true,\"score_out_of_100\":100,\"additional_feedback\":\"The submitted answer is a superset of the expert answer and is fully consistent with it.\",\"factual_consistency\":\"B\"}"
            }
          }
        ]
      },
      "logprobs": null,
      "finish_reason": "tool_calls"
    }
  ],
  "usage": {
    "prompt_tokens": 790,
    "completion_tokens": 53,
    "total_tokens": 843,
    "total_cost": 0.0004745
  }
}