 # We have a data set (n=100) with hormone concentrations and the frequency of each value. Assuming normal (continuous) distributions, We need to determine the parameters of the continuous probability distribution functions. Cheyanne Leigh 2020-11-26 Answered
We have a data set (n=100) with hormone concentrations and the frequency of each value. Assuming normal (continuous) distributions, We need to determine the parameters of the continuous probability distribution functions.
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Step1
Let us suppose ${X}_{1},{X}_{2},\dots .,{X}_{100}$ are samples from normal distribution
with mean $\mu \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}variance{\sigma }^{2}$
Hence, ${X}_{i}\sim N\left(\mu ,{\sigma }^{2}\right)$
Here note that $\mu \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\sigma }^{2}$ are the parameters of the probability distribution
Now, You have calculated mean, Variance, and standard deviation of the samples.
Sample mean,$\stackrel{―}{X}=\frac{1}{100}\sum _{i}{X}_{i}$,
sample variance , ${S}^{2}=\frac{1}{100-1}\sum _{i=1}^{100}{\left({X}_{i}-\stackrel{―}{x}\right)}^{2}Now,E\left(\stackrel{―}{x}\right)=\frac{1}{100}$
$\sum _{i}\left({X}_{i}\right)=\frac{1}{100}×100×\mu =\mu$
and it can be shown $E\left({S}^{2}\right)={\sigma }^{2}$
Step 2
Hence, the sample mean and sample variance are the estimates of the population parameter $\mu \phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\sigma }^{2}$
The parameters of the probability distribution will be sample mean that you have calculated and sample variance that you have calculated.

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