Basic Multivariate Flashcards by O n

What is a population vector?

A population vector is a mathematical representation of a population’s characteristics, often used in statistical analysis.

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True or False: The population mean vector is the average of all elements in a population.

True

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Fill in the blank: The variance matrix is a square matrix that contains the _____ of each variable along its diagonal.

variance

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What does the covariance matrix represent?

The covariance matrix represents the covariance between pairs of variables in a dataset.

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Multiple Choice: Which of the following matrices shows the strength and direction of a linear relationship between variables? A) Variance matrix B) Covariance matrix C) Correlation matrix D) Population mean vector

C) Correlation matrix

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What is multivariate skewness?

Multivariate skewness is a measure of the asymmetry of the probability distribution of a multivariate dataset.

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True or False: Multivariate kurtosis measures the ‘tailedness’ of a multivariate distribution.

True

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Fill in the blank: The formula for multivariate skewness involves the _____ matrix.

covariance

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What is the purpose of the sample correlation matrix?

The sample correlation matrix quantifies the linear relationship between multiple variables in a dataset.

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Multiple Choice: Which of the following is a common formula for calculating sample correlation?

Pearson correlation coefficient

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What are the primary components used in the calculation of kurtosis?

The fourth central moment and the variance of the dataset.

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True or False: A multivariate normal distribution has a skewness of zero.

True

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What does a high value of multivariate kurtosis indicate?

It indicates a distribution with heavy tails or outliers.

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Fill in the blank: The sample correlation matrix is derived from the _____ of the variables.

covariance

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Short Answer: How do you interpret a sample correlation coefficient of 0.8?

It indicates a strong positive linear relationship between the two variables.

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What is a multivariate normal distribution?

A multivariate normal distribution is a generalization of the one-dimensional normal distribution to higher dimensions, characterized by a mean vector and a covariance matrix.

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True or False: The linear combination of independent normal variables is also normally distributed.

True

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Fill in the blank: The mean of a linear combination of random variables is equal to the __________ of the means of the individual variables.

linear combination

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What does the covariance matrix represent in a multivariate normal distribution?

The covariance matrix represents the variances and covariances between the different dimensions of the multivariate distribution.

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If Y is a multivariate normal random vector, what is the distribution of AY + b for any matrix A and vector b?

AY + b is also multivariate normal.

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What is the effect of a linear transformation on the mean of a multivariate normal distribution?

The mean of the transformed distribution is given by the linear transformation applied to the original mean.

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True or False: The sum of two independent multivariate normal variables is multivariate normal.

True

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What is the relationship between the variances of a linear combination of independent variables?

The variance of the linear combination is the sum of the variances of the individual variables, weighted by the square of their coefficients.

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Define the term 'linear combination' in the context of random variables.

A linear combination of random variables is an expression formed by multiplying each variable by a constant and adding the results together.

What is the formula for the variance of a linear combination of two random variables X and Y, given weights a and b?

Var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X, Y)

Fill in the blank: The joint distribution of a multivariate normal vector is completely described by its __________ and __________.

mean vector, covariance matrix

What is the characteristic function of a multivariate normal distribution?

The characteristic function of a multivariate normal distribution is given by exp(iμ^T t - 0.5 t^T Σ t), where μ is the mean vector and Σ is the covariance matrix.

True or False: The marginal distribution of any subset of a multivariate normal distribution is also normal.

True

What happens to the shape of the distribution when the covariance matrix is singular?

The distribution becomes degenerate, meaning it does not cover the full space and collapses onto a lower-dimensional subspace.

What is the significance of the determinant of the covariance matrix in a multivariate normal distribution?

The determinant of the covariance matrix indicates the volume of the distribution in the multivariate space; a zero determinant means no variation in one or more dimensions.

How does one compute the moment-generating function of a multivariate normal distribution?

The moment-generating function is computed using the formula M(t) = exp(μ^T t + 0.5 t^T Σ t), where μ is the mean vector and Σ is the covariance matrix.

What is the role of the correlation coefficient in a multivariate normal distribution?

The correlation coefficient quantifies the degree of linear relationship between two variables in the distribution.

Fill in the blank: The property of __________ states that any linear combination of independent normal random variables is normally distributed.

closure

What does it mean for two random variables to be uncorrelated in a multivariate normal distribution?

Uncorrelated random variables have a covariance of zero, indicating that knowing the value of one does not provide information about the other.

What is the implication of a multivariate normal distribution having a diagonal covariance matrix?

A diagonal covariance matrix implies that the variables are uncorrelated.

True or False: The linear combination of dependent normal variables is not normally distributed.

False

What does the term 'independence' imply in the context of multivariate normal distributions?

Independence implies that the joint distribution can be expressed as the product of the marginal distributions.

What is the geometric interpretation of the contours of a bivariate normal distribution?

The contours of a bivariate normal distribution are ellipses centered around the mean, with axes determined by the variances and covariances.

Fill in the blank: The __________ of a multivariate normal distribution is always symmetric and positive semi-definite.

covariance matrix