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.

Question
Data distributions
asked 2020-11-26
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.

Answers (1)

2020-11-27
Step1
Let us suppose \(\displaystyle{X}_{{1}},{X}_{{2}},\ldots.,{X}_{{100}}\) are samples from normal distribution
with mean \(\displaystyle\mu{\quad\text{and}\quad}{v}{a}{r}{i}{a}{n}{c}{e}\sigma^{{2}}\)
Hence, \(\displaystyle{X}_{{i}}\sim{N}{\left(\mu,\sigma^{{2}}\right)}\)
Here note that \(\displaystyle\mu{\quad\text{and}\quad}\sigma^{{2}}\) are the parameters of the probability distribution
Now, You have calculated mean, Variance, and standard deviation of the samples.
Sample mean,\(\displaystyle\overline{{X}}=\frac{{1}}{{100}}\sum_{{i}}{X}_{{i}}\),
sample variance , \(\displaystyle{S}^{{2}}=\frac{{1}}{{{100}-{1}}}{\sum_{{{i}={1}}}^{{100}}}{\left({X}_{{i}}-\overline{{x}}\right)}^{{2}}{N}{o}{w},{E}{\left(\overline{{x}}\right)}=\frac{{1}}{{100}}\)
\(\displaystyle\sum_{{i}}{\left({X}_{{i}}\right)}=\frac{{1}}{{100}}\times{100}\times\mu=\mu\)
and it can be shown \(\displaystyle{E}{\left({S}^{{2}}\right)}=\sigma^{{2}}\)
Step 2
Hence, the sample mean and sample variance are the estimates of the population parameter \(\displaystyle\mu{\quad\text{and}\quad}\sigma^{{2}}\)
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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