给定 1 维正态分布  𝑁 ( 𝜇 , 𝜎 2 )    中的  𝑛   个i.i.d 样本 𝑥 𝑖   。如果 对  𝜇   和   𝜎  使用 MLE，下列哪一项是正确的似然函数？   Given 𝑛   i.i.d samples 𝑥 𝑖   from the 1-D normal distribution 𝑁 ( 𝜇 , 𝜎 2 ) . If using MLE for  𝜇 and   𝜎  . Which of the following is the right likelihood function?单项选择题

A

𝐿 ( 𝜇 , 𝜎 ) = 𝑝 ( 𝐷 | 𝜇 , 𝜎 ) = ( 1 𝜎 2 𝜋 ) ∏ 𝑖 = 1 𝑛 𝑒 − ( 𝑥 𝑖 − 𝜇 ) 2 2 𝜎 2

B

𝐿 ( 𝜇 , 𝜎 ) = 𝑝 ( 𝐷 | 𝜇 , 𝜎 ) = ( 1 𝜎 2 𝜋 ) 𝑛 ∑ 𝑖 = 1 𝑛 𝑒 ( 𝑥 𝑖 − 𝜇 ) 2 2 𝜎 2

C

𝐿 ( 𝜇 , 𝜎 ) = 𝑝 ( 𝐷 | 𝜇 , 𝜎 ) = ( 1 𝜎 2 𝜋 ) 𝑛 ∏ 𝑖 = 1 𝑛 𝑒 − ( 𝑥 𝑖 − 𝜇 ) 2 2 𝜎 2

D

𝐿 ( 𝜇 , 𝜎 ) = 𝑝 ( 𝐷 | 𝜇 , 𝜎 ) = ( 1 2 𝜋 ) 𝑛 ∏ 𝑖 = 1 𝑛 𝑒 − ( 𝑥 𝑖 − 𝜇 ) 2 2 𝜎 2

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类似问题

Suppose X follows a distribution with two parameters, and we want to estimate both via MLE.  Once we have the log-likelihood function ℓ ( 𝜃 1 , 𝜃 2 ) , what should we do next?

Suppose  𝑋 ∼ 𝐵 𝑖 𝑛 𝑜 𝑚 ( 𝑛 = 10 , 𝑝 ) and we observe a random sample of size 3:  𝑥 1 = 0 , 𝑥 2 = 3 , and  𝑥 3 = 1 .  Once we have the log-likelihood function ℓ ( 𝑝 ) , what value will be 𝑝 ^ , the MLE for 𝑝 ?

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假设从正态分布中抽取  5  个样本给你， {0, 3, 1, -1, 2} ，我们想使用最大似然估计来估计方差。下列哪一项最接近解？   Suppose you are given 5 samples that are drawn from a normal distribution, {0, 3, 1, -1, 2}, and we want to use maximum likelihood estimation for estimating the variance. Which of the following is closest to the solution?

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