Question

Suppose \(Y \sim N_{p}(\mu, \Omega)\) and \(a\) and \(b\) are \(p \times 1\) vectors of constants. Find the distribution of \(X_{1}=a^{\mathrm{T}} Y\) conditional on \(X_{2}=b^{\mathrm{T}} Y=x_{2} .\) Under what circumstances does this not depend on \(x_{2} ?\)

Short answer

The conditional distribution is Gaussian with specified mean and variance. It is independent of \(x_2\) if \(a^T \Omega b = 0\).

Step-by-step solution

Define the Joint Distribution

Given that \( Y \sim N_p(\mu, \Omega) \), we begin by noting that the linear transformations \( X_1 = a^T Y \) and \( X_2 = b^T Y \) are linear combinations of a normally distributed vector \( Y \). Therefore, the joint distribution of \( (X_1, X_2) \) is also normally distributed, because any linear transformation of a multivariate Gaussian remains Gaussian.

Write the Linear Transformations

Let \( Y = \mu + Z \), where \( Z \sim N_p(0, \Omega) \). The linear transformations are specified as \( X_1 = a^T Y = a^T \mu + a^T Z \) and \( X_2 = b^T Y = b^T \mu + b^T Z \). This shows that both \( X_1 \) and \( X_2 \) are linear combinations of \( Y \), which will help us find the joint distribution.

Express the Mean and Covariance of the Joint Distribution

The mean of the joint distribution \( (X_1, X_2) \) is \( (a^T \mu, b^T \mu) \). The covariance matrix is given by \[ \text{Cov}(X_1, X_2) = \begin{pmatrix} a^T \Omega a & a^T \Omega b \ b^T \Omega a & b^T \Omega b \end{pmatrix}. \] This matrix shows the relationship and variance between \( X_1 \) and \( X_2 \).

Conditional Distribution of \( X_1 \) Given \( X_2 = x_2 \)

To find the distribution of \( X_1 \) conditional on \( X_2 = x_2 \), we use the properties of the multivariate normal distribution. The conditional distribution also results in a Gaussian distribution: - The conditional mean is given by \[ a^T \mu + \frac{a^T \Omega b}{b^T \Omega b}(x_2 - b^T \mu). \]- The conditional variance is \[ a^T \Omega a - \frac{(a^T \Omega b)^2}{b^T \Omega b}. \] Therefore, \( X_1 | X_2 = x_2 \sim N\left(a^T \mu + \frac{a^T \Omega b}{b^T \Omega b}(x_2 - b^T \mu), a^T \Omega a - \frac{(a^T \Omega b)^2}{b^T \Omega b}\right). \)

Condition for Independence from \( x_2 \)

The conditional distribution mean \[ a^T \mu + \frac{a^T \Omega b}{b^T \Omega b}(x_2 - b^T \mu) \] does not depend on \( x_2 \) if \( \frac{a^T \Omega b}{b^T \Omega b} = 0 \). This occurs when \( a^T \Omega b = 0 \), meaning \( X_1 \) and \( X_2 \) are uncorrelated. Since they are Gaussian, being uncorrelated implies independence.

Key concepts

Conditional Distribution

In the realm of statistics, understanding conditional distribution is essential for analyzing how a random variable behaves given specific conditions. In the problem at hand, you have two linear transformations of a normally distributed multivariate vector, namely, \( X_1 = a^T Y \) and \( X_2 = b^T Y \). The conditional distribution of \( X_1 \) given \( X_2 = x_2 \) involves determining how the knowledge of \( X_2 \)'s specific outcome (or value) affects the behavior of \( X_1 \).

When a multivariate normal distribution is involved, such as in this example, the conditional distribution remains Gaussian. This is a powerful property that simplifies many real-world problems, allowing analysts to isolate and understand the contributions of specific variables within a multivariate framework.

• The conditional mean adjusts for any known outcomes of \( X_2 \) to better predict \( X_1 \).
• The conditional variance reveals the degree of variability in \( X_1 \) after considering \( X_2 \).

This exploration helps us predict and understand scenarios involving dependencies and influences within a dataset.

Linear Transformation

Linear transformation is a central concept in linear algebra and statistics, often used to change or simplify data representations. In this context, the transformation involves vectors \( a \) and \( b \), which are applied to the random vector \( Y \) to derive \( X_1 \) and \( X_2 \).

This transformation leverages the property that any linear combination of a normally distributed random vector results in another normally distributed variable. The notation \( X_1 = a^T Y \) and \( X_2 = b^T Y \) represents these transformations.

Linear transformations preserve the distributional form (i.e., normality) when applied to Gaussian variables. This ensures that even after transformation, we can apply further analyses using properties of normal distributions.

• Transforms complex relationships into simpler, more interpretable forms.
• Enables easier computation of statistics such as means and variances.
• Critical for understanding dependencies and influences in multivariate data.

Linear transformation is thus a versatile tool that provides insightful views into data behavior and relationships.

Covariance Matrix

The covariance matrix is a key concept when dealing with multivariate distributions. It provides insights into how variables in a dataset interact and vary together. In the case of our particular problem, the covariance matrix for \( (X_1, X_2) \) is crucial in determining their relationship.

The covariance matrix in this context is specified as:\[\text{Cov}(X_1, X_2) = \begin{pmatrix} a^T \Omega a & a^T \Omega b \ b^T \Omega a & b^T \Omega b \end{pmatrix}.\]
This matrix uncovers the interrelationship between \( X_1 \) and \( X_2 \):

• The diagonal terms (\( a^T \Omega a \) and \( b^T \Omega b \)) denote the variances of \( X_1 \) and \( X_2 \), respectively.
• The off-diagonal terms (\( a^T \Omega b \) and \( b^T \Omega a \)) reveal the covariance, or how \( X_1 \) and \( X_2 \) change together.

In essence, the covariance matrix provides a comprehensive picture of the relationships and variability between the components of a joint distribution.

Gaussian Distribution

The Gaussian distribution, also known as the normal distribution, is pivotal in statistics due to its natural occurrence in various phenomena and its properties around mean and variance. When a vector like \( Y \) is normally distributed, any linear transformations remain Gaussian due to mathematical properties of normal distributions.

In the given problem, both \( X_1 \) and \( X_2 \) result from linear transformations of \( Y \) and therefore are normally distributed. Through these transformations, the joint distribution of \( (X_1, X_2) \) remains Gaussian, ensuring that all subsequent analyses remain tractable using known properties of normal distributions.

Highlights of Gaussian distribution involvement include:

• Mean and variance fully describe the distribution, simplifying predictions and insights.
• Linear transformations preserve normality, which aids in modeling real-world systems.

This characteristic makes Gaussian distributions extremely valuable in statistical analysis and probability theory.

Uncorrelated Variables

Uncorrelated variables are those between which no statistical relationship can be identified using covariance. In the context of this problem, \( X_1 \) and \( X_2 \) are uncorrelated if their covariance \( a^T \Omega b \) equals zero.

For Gaussian variables like \( X_1 \) and \( X_2 \), being uncorrelated also means they are independent due to their distribution properties. This indicates that knowing \( X_2 \) does not provide any information about \( X_1 \), and vice versa.

The situation where the conditional mean of \( X_1 \) does not depend on \( x_2 \) further illustrates this independence. When the covariance is zero:

• There is no direct influence or predictable relationship between the variables.
• Each variable's behavior is solely determined by its own distribution without interference.

In sum, understanding when variables are uncorrelated helps in identifying independence, simplifying analysis, and making robust predictions.