A question about 'properties of multivariate normal distributions'.

in summary, the chi-square distribution is the sum of the squares of p independent standard normal distributions, and this is a $\chi_p^2$ distribution.
  • #1
rof
4
0
hello guys,
i have a question and looking for an answer quickly...

question;

prove that,
XxZdUxc.png

Z is a vector.thank you.
 
Physics news on Phys.org
  • #2
rof said:
hello guys,
i have a question and looking for an answer quickly...

question;

prove that,

Z is a vector.thank you.

Hi rof! Welcome to MHB! ;)

The chi-square distribution is defined as the sum of the squares of p independent standard normal distributions.
So your proof follows from the definition.
 
  • #3
this is the full question paper. (if that helps you to understand the nature of the question)
the answer i want is to question number 5.

View attachment 5906

thank you
 

Attachments

  • QnBxP5g.jpg
    QnBxP5g.jpg
    89.9 KB · Views: 60
Last edited by a moderator:
  • #4
rof said:
this is the full question paper. (if that helps you to understand the nature of the question)
the answer i want is to question number 5.http://i.imgur.com/l2CpJz3.jpgthank you

That link is broken...and rather than using a free online image hosting service (many of which don't work well), it is better to attach images inline so that everyone can see them and they are not subject to the image hosting service taking them down.
 
  • #5
MarkFL said:
That link is broken...and rather than using a free online image hosting service (many of which don't work well), it is better to attach images inline so that everyone can see them and they are not subject to the image hosting service taking them down.

i corrected it by editing the post as soon as i posted it, but somehow editing didnt happen.

here is the link;
http://i.imgur.com/QnBxP5g.jpg

5th question
 
  • #6
rof said:
i corrected it by editing the post as soon as i posted it, but somehow editing didnt happen.

here is the link;
http://i.imgur.com/QnBxP5g.jpg

5th question

I went ahead and edited your previous post to attach the image inline. (Yes)
 
  • #7
rof said:
this is the full question paper. (if that helps you to understand the nature of the question)
the answer i want is to question number 5.

thank you

Your vector $\mathbf Z$ consists of $p$ independent standard normal distributions.
When we calculate the dot product $\mathbf Z' \mathbf Z$, we get the sum of the squares of those $p$ independent standard normal distributions.
That is a $\chi_p^2$ distribution by definition.
 
  • #8
thanks all for taking time to read my question and providing answers.
good luck to all... :)
 

What is a multivariate normal distribution?

A multivariate normal distribution is a probability distribution that describes the behavior of a set of multiple random variables that are normally distributed. It is characterized by its mean vector, which represents the average value of each variable, and its covariance matrix, which describes the relationship between each variable.

What are the properties of a multivariate normal distribution?

The properties of a multivariate normal distribution include:

  • The variables are normally distributed.
  • The mean vector and covariance matrix uniquely determine the distribution.
  • The variables are independent if and only if the covariance matrix is diagonal.
  • The variables are uncorrelated if and only if the covariance matrix is zero.
  • The variables are jointly normal.

How is a multivariate normal distribution different from a univariate normal distribution?

A univariate normal distribution describes the behavior of a single random variable, while a multivariate normal distribution describes the behavior of a set of multiple random variables. In a univariate normal distribution, there is only one mean and one variance, while in a multivariate normal distribution, there is a mean vector and a covariance matrix.

What are some real-world applications of multivariate normal distributions?

Multivariate normal distributions are commonly used in statistical analysis and modeling in various fields, including finance, economics, psychology, and biology. They are particularly useful for analyzing data sets with multiple variables, such as stock prices, consumer behaviors, and brain activity.

How can I determine if my data follows a multivariate normal distribution?

There are several statistical tests that can be used to determine if a data set follows a multivariate normal distribution, such as the Shapiro-Wilk test or the Kolmogorov-Smirnov test. Additionally, visual inspection of the data using histograms or Q-Q plots can also provide insight into the distribution of the data. It is important to note that these tests are not definitive and should be used in conjunction with other analyses to make a determination.

Similar threads

I Variable transformation for a multivariate normal distribution
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
I Citation needed: Only multivariate rotationally invariant distribution with iid components is a multivariate normal distribution
    • Set Theory, Logic, Probability, Statistics
Replies
3
Views
118
I Proving a multivariate normal distribution by the moment generating function
    • Set Theory, Logic, Probability, Statistics
Replies
5
Views
2K
I Normality of errors and residuals in ordinary linear regression
    • Set Theory, Logic, Probability, Statistics
Replies
3
Views
481
A A different discrete normal distribution
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
I Relating Moments from one Distribution to the Moments of Another
    • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K
I Distribution of Sum of Two Weird Random Variables....
    • Set Theory, Logic, Probability, Statistics
Replies
5
Views
1K
I Linear regression and random variables
    • Set Theory, Logic, Probability, Statistics
Replies
30
Views
2K
I Desirable estimator properties...
    • Set Theory, Logic, Probability, Statistics
Replies
7
Views
479
A The distribution that has a certain distribution as its limit case
    • Set Theory, Logic, Probability, Statistics
Replies
7
Views
1K

Hot Threads

Recent Insights

Back
Top