gpt现在能证明欧式几何证明题吗，上网搜了下，没看见实例或者证据



【 在 fanci 的大作中提到: 】
这是怎么做到的，是不是连接了符号计算库，不然仅靠阅读数学课本很难提炼出不定积分能力吧？

问：Let X be a standard normal random variable. What is the expectation of the absolute value of X?

答：The expectation of the absolute value of a standard normal random variable X can be computed as follows:

E[|X|] = ∫_{-∞}^{∞} |x| * f(x) dx

where f(x) is the probability density function of the standard normal distribution, given by:

f(x) = (1 / sqrt(2π)) * exp(-x^2 / 2)

Using the symmetry properties of the standard normal distribution, we can write:

E[|X|] = 2 * ∫_{0}^{∞} x * f(x) dx

Substituting the expression for f(x) and simplifying, we obtain:

E[|X|] = 2 * ∫_{0}^{∞} x * (1 / sqrt(2π)) * exp(-x^2 / 2) dx

This integral can be evaluated using integration by substitution, letting u = x^2 / 2 and du = x dx, and simplifying:

E[|X|] = 2 * ∫_{0}^{∞} sqrt(2 / π) * exp(-u) du

Using the fact that ∫_{0}^{∞} exp(-u) du = 1, we can simplify further:

E[|X|] = 2 * sqrt(2 / π) * ∫_{0}^{∞} exp(-u) du

E[|X|] = 2 * sqrt(2 / π)

Therefore, the expectation of the absolute value of a standard normal random variable X is 2 * sqrt(2 / π), which is approximately equal to 1.1284.
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