How can you estimate the parameters of a normal distribution?

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How can you estimate the parameters of a normal distribution?

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Answer 1 · 2016-01-22T19:36:21

Use the sample mean $¯ x$ $bar{x}$ and the sample standard deviation $s$ $s$ to estimate the mean $μ$ $mu$ and standard deviation $σ$ $sigma$ of the normal distribution you wish to use.

Explanation:

The normal distribution has probability density function (pdf) $f (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{{(x - μ)}^{2}}{2 σ^{2}}}$ $f(x)=1/(sigma sqrt{2pi})e^{-(x-mu)^2/(2sigma^2)}$ . The parameter $μ$ $mu$ is its mean and the parameter $σ$ $sigma$ is its standard deviation.

If you have data from a random sample and compute the sample mean $¯ x = \frac{x_{1} + x_{2} + x_{3} + \dots + x_{n}}{n}$ $bar{x}=(x_1+x_2+x_3+cdots+x_n)/n$ and the sample standard deviation $s = \sqrt{\frac{{(x_{1} - ¯ x)}_{1}^{2} + {(x_{2} - ¯ x)}_{2}^{2} + \dots + {(x_{n} - ¯ x)}_{n}^{2}}{n - 1}}$ $s=sqrt{((x_1-bar{x})^2+(x_2-bar{x})^2+cdots+(x_n-bar{x})^2)/(n-1)}$ , these will be good estimates for $μ$ $mu$ and $σ$ $sigma$ when your sample size $n$ $n$ is sufficient large.

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How can you estimate the parameters of a normal distribution?

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