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Q 8.2. Suppose that

X_{1}

and

X_{2}

are two random variables whose joint di...

Nov 30, 2024

Solution by Steps

step 1

To construct the random variables

Z_{1}

and

Z_{2}

, we can use the following transformations: let

Z_{1} = X_{1}

and

Z_{2} = \frac{X_{2} - \rho Z_{1}}{\sqrt{1 - \rho^2}}

. This ensures that

Z_{1}

and

Z_{2}

are uncorrelated

step 2

We need to show that

Z_{1}

and

Z_{2}

have a joint standard Gaussian distribution. Since

X_{1}

and

X_{2}

are jointly Gaussian, any linear transformation of them will also be jointly Gaussian

step 3

The means of

Z_{1}

and

Z_{2}

are both zero:

\mathbf{E}[Z_{1}] = \mathbf{E}[X_{1}] = 0

and

\mathbf{E}[Z_{2}] = \mathbf{E}\left[\frac{X_{2} - \rho X_{1}}{\sqrt{1 - \rho^2}}\right] = 0

step 4

The variances are

\mathbf{E}[Z_{1}^2] = \mathbf{E}[X_{1}^2] = 1

and

\mathbf{E}[Z_{2}^2] = \mathbf{E}\left[\left(\frac{X_{2} - \rho X_{1}}{\sqrt{1 - \rho^2}}\right)^2\right] = 1

. Thus, both

Z_{1}

and

Z_{2}

have unit variance

step 5

The covariance

\mathbf{E}[Z_{1}Z_{2}] = \mathbf{E}[X_{1} \cdot \frac{X_{2} - \rho X_{1}}{\sqrt{1 - \rho^2}}] = \frac{\mathbf{E}[X_{1}X_{2}] - \rho \mathbf{E}[X_{1}^2]}{\sqrt{1 - \rho^2}} = \frac{\rho - \rho}{\sqrt{1 - \rho^2}} = 0

. Thus,

Z_{1}

and

Z_{2}

are uncorrelated

step 6

Since

Z_{1}

and

Z_{2}

are jointly Gaussian with zero means, unit variances, and zero covariance, they follow the standard Gaussian distribution on

\mathbb{R}^{2}

Answer

Z_{1}

and

Z_{2}

are standard Gaussian random variables.

Key Concept

Jointly Gaussian distributions and linear transformations.

Explanation

The construction of

Z_{1}

and

Z_{2}

from

X_{1}

and

X_{2}

ensures they are uncorrelated and jointly Gaussian, satisfying the properties of the standard Gaussian distribution.