Multivariate Normal Distribution

A^-1 .)Finally, don't forget to specify the dimension of the new distribution (i.e. give its mean vector and covariance matrix).In summary, the distribution of Y is normal, with a mean vector of A^-1*0 = 0 and a covariance matrix of (A^-1)^T * E * A^-1 = (A^-1)^T * A * A^-1 = I.
  • #1
gatorain
1
0

Homework Statement


Z = (Z1, Z2, ... Zd) is a d-dimensional normal variable with distribution N(0, E).

Let A be invertible matrix such that AA' = E. (E = sigma = covariance matrix).

Find the distribution of Y = (A^-1)*Z.

The Attempt at a Solution



I'm pretty sure the solution is normal, but what would be its mean and variance?
 
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  • #2
A couple of hints.

1. What are the definitions of the mean and variance (you omitted the Relevant equations)? Hint: They involve integrals.

2. The matrix A, and thus its inverse, is a constant. Hint: You can take constants outside of the integral.
 
  • #3
You don't need integration for this if you know how the mean vector and covariance matrix for multivariate distributions work.

If [tex] Y [/tex] is a random vector with mean vector [tex] \mu [/tex], the mean of [tex] A Y [/tex] is

[tex]
E(A Y)
[/tex]

How can you simplify that? (This may be what the other poster meant by "integration" - if so, I apologize)

The covariance matrix of [tex] AY [/tex] can also be easily simplified. (Hint: this is where you'll use your fact about [tex] A [/tex]
 

Related to Multivariate Normal Distribution

What is the Multivariate Normal Distribution?

The Multivariate Normal Distribution is a probability distribution that is used to model multivariate data, meaning data with more than one variable. It is a generalization of the normal distribution, which is used to model single variable data.

What are the characteristics of the Multivariate Normal Distribution?

The Multivariate Normal Distribution is characterized by its mean vector and covariance matrix. The mean vector represents the average values of each variable, and the covariance matrix represents the relationships between each variable.

How is the Multivariate Normal Distribution visualized?

The Multivariate Normal Distribution is typically visualized as a bell-shaped curve in multiple dimensions. For example, in two dimensions, it would look like an elliptical shape. This curve represents the probability of different combinations of values for each variable.

What is the importance of the Multivariate Normal Distribution in statistics?

The Multivariate Normal Distribution is important in statistics because it is used in many statistical methods, such as hypothesis testing, regression analysis, and clustering. It also allows for the analysis of multivariate data, which is often more complex and realistic than single variable data.

What are some real-world applications of the Multivariate Normal Distribution?

The Multivariate Normal Distribution is used in many fields, including finance, biology, and psychology. Some specific applications include portfolio optimization in finance, multivariate analysis of genetic data in biology, and modeling personality traits in psychology.

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