# Statistics, calculate the distribution

• MaxManus
In summary, the goal is to find a function g that does not depend on μ, and the function g is found to be the pdf of the sample mean.
MaxManus

## Homework Statement

Assume $z_1, ..., z_m$ are iid,$z_i = μ+\epsilon_i$

$\epsilon_i]$is N(0,σ^2)

Show that
f(z; μ) = g($\bar{z}$; μ)h(z)
where h(·) is a function not depending on μ.

## The Attempt at a Solution

Now z is normal distributed with mean my and variance sigma^2

$f(z,\mu) = \frac{1}{\sigma^2 \sqrt{2 \pi}} e^{-\frac{(z-\mu)^2}{2 \sigma^2}}$

f(z; μ) = $\prod_{i=1}^m \frac{1}{\sigma^2 \sqrt{2 \pi}} e^{-\frac{(z_i-\mu)^2}{2 \sigma^2}}$

$f(\bf{z},\mu) = (\frac{1}{\sigma^2 \sqrt{2 \pi}})^m \prod_{i=1}^m e^{-\frac{(z_i-\mu)^2}{2 \sigma^2}}$

but how do I go from here to
f(z; μ) = g($\bar{z}$; μ)h(z)

And am I in the right track?

Last edited:
MaxManus said:

## Homework Statement

Assume $z_1, ..., z_m$ are iid,$z_i = μ+\epsilon_i$

$\epsilon_i]$is N(0,σ^2)

Show that
f(z; μ) = g($\bar{z}$; μ)h(z)
where h(·) is a function not depending on μ.

## The Attempt at a Solution

Now z is normal distributed with mean my and variance sigma^2

$f(z,\mu) = \frac{1}{\sigma^2 \sqrt{2 \pi}} e^{-\frac{(z-\mu)^2}{2 \sigma^2}}$

f(z; μ) = $\prod_{i=1}^m \frac{1}{\sigma^2 \sqrt{2 \pi}} e^{-\frac{(z_i-\mu)^2}{2 \sigma^2}}$

$f(\bf{z},\mu) (\frac{1}{\sigma^2 \sqrt{2 \pi}})^m \prod_{i=1}^m e^{-\frac{(z_i-\mu)^2}{2 \sigma^2}}$

but how do I go from here to
f(z; μ) = g($\bar{z}$; μ)h(z)

And am I in the right track?

I assume the notation means that $g(\bar{z})$ is the pdf of the sample mean. If so, you need to find the function g. What properties of the normal distribution are important for doing that? What is the importance of the fact that the $z_i$ are independent?

RGV

Thanks

I'm not sure what the notation mean, but I assumed g was a function of the sample mean and mu. But if it is the pdf of the sample mean.
The pdf of $\bar{z}$ is the pdf to the normal distribution with mean $\mu]$ and variance $\sigma^2/m$
So
$g(\bar{z}) = \frac{1}{(\sigma^2/m )\sqrt{2 \pi}} e^{-\frac{(\bar{z}-\mu)^2}{2 (\sigma^2/m)}}$

But I'm not sure where to go from here.

Last edited:
MaxManus said:
Thanks

I'm not sure what the notation mean, but I assumed g was a function of the sample mean and mu. But if it is the pdf of the sample mean.
The pdf of $\bar{z}$ is the pdf to the normal distribution with mean $\mu]$ and variance $\sigma^2/m$
So
$g(\bar{z}) = \frac{1}{(\sigma^2/m )\sqrt{2 \pi}} e^{-\frac{(\bar{z}-\mu)^2}{2 (\sigma^2/m)}}$

But I'm not sure where to go from here.

So,
$$g = \frac{1}{\sqrt{2\pi}\sigma/\sqrt{n}}\exp{\left[\left(\frac{z_1+z_2+\cdots+z_n}{n}-\mu \right)^2/(2 \sigma^2 /n)\right]}.$$

You are supposed to show that f(z,μ)/g does not have μ in it.

PS: your expressions for the normal distributions are a bit wrong: you should have$\frac{1}{\sigma \sqrt{2 \pi}}, \text{ not } \frac{1}{\sigma^2 \sqrt{2 \pi}}$ in front.

RGV

Thanks for all the help.

To those who want to see the rest of the calculation it is in the attachment.

#### Attachments

• IMG_0001.pdf
308.8 KB · Views: 200

## What is the purpose of statistical distribution?

The purpose of statistical distribution is to describe the frequency and pattern of occurrence of a particular variable or set of data. It allows us to understand the relationship between different variables and make predictions about future outcomes.

## What are the different types of statistical distributions?

There are many types of statistical distributions, but some of the most commonly used ones include normal distribution, binomial distribution, Poisson distribution, and exponential distribution. Each distribution has its own characteristics and is used for different types of data analysis.

## How do you calculate the distribution of a data set?

To calculate the distribution of a data set, you first need to organize the data into a frequency table or histogram. Then, you can use mathematical formulas or statistical software to calculate the mean, median, mode, and other measures of central tendency. You can also plot the data on a graph to visualize the distribution.

## What is the importance of understanding statistical distribution?

Understanding statistical distribution is crucial for making informed decisions and drawing accurate conclusions from data. It helps identify patterns, trends, and outliers in the data, which can provide valuable insights for businesses, researchers, and policymakers.

## How is statistical distribution used in real life?

Statistical distribution is used in various fields, including finance, healthcare, marketing, and social sciences. For example, it helps investors analyze stock market trends, doctors determine the effectiveness of a new treatment, and marketers identify customer preferences. It is also used in quality control to ensure that products meet certain standards.

• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
2K
• Calculus and Beyond Homework Help
Replies
3
Views
678
• Calculus and Beyond Homework Help
Replies
1
Views
268
• Calculus and Beyond Homework Help
Replies
6
Views
969
• Calculus and Beyond Homework Help
Replies
3
Views
818
• Calculus and Beyond Homework Help
Replies
1
Views
997
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
487