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Gaussian Random Vectors

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1: Let

X \sim N(\mu_X, \Sigma_X)

be a bivariate Gaussian random vector, where

\mu_X = \begin{bmatrix} 0 \\ 1 \end{bmatrix}

and

\Sigma_X = \begin{bmatrix} 1 & 0.5 \\ 0.5 & 2 \end{bmatrix}

. Find the conditional distribution of

X_1

given

X_2 = 1

.

a:

X_1 | X_2 = 1 \sim N(0.25, 0.75)

Explanation

Explanation

b:

X_1 | X_2 = 1 \sim N(0.5, 1)

Explanation

Explanation

c:

X_1 | X_2 = 1 \sim N(0.25, 0.5)

Explanation

Explanation

d:

X_1 | X_2 = 1 \sim N(0.75, 0.75)

Explanation

Explanation

2: Suppose

X \sim N(\mu_X, \Sigma_X)

is a Gaussian random vector in

\mathbb{R}^2

, where

\Sigma_X = \begin{bmatrix} 1 & 0.7 \\ 0.7 & 1 \end{bmatrix}

. What is the correlation coefficient

\rho(X_1, X_2)

?

a:

\rho(X_1, X_2) = 0.7

Explanation

Explanation

b:

\rho(X_1, X_2) = 1.0

Explanation

Explanation

c:

\rho(X_1, X_2) = 0.49

Explanation

Explanation

d:

\rho(X_1, X_2) = 0.5

Explanation

Explanation

3: Given the cumulative distribution function (CDF) of the standard normal distribution, what is the value of

\lim_{x \to \infty} \Phi(x)

and

\lim_{x \to -\infty} \Phi(x)

?

a:

\lim_{x \to \infty} \Phi(x) = 1

,

\lim_{x \to -\infty} \Phi(x) = 0

Explanation

Explanation

b:

\lim_{x \to \infty} \Phi(x) = 0

,

\lim_{x \to -\infty} \Phi(x) = 1

Explanation

Explanation

c:

\lim_{x \to \infty} \Phi(x) = 0.5

,

\lim_{x \to -\infty} \Phi(x) = 0.5

Explanation

Explanation

d:

\lim_{x \to \infty} \Phi(x) = 2

,

\lim_{x \to -\infty} \Phi(x) = -1

Explanation

Explanation

4: What is the value of the CDF of the standard normal distribution at

x = 0

?

a:

\Phi(0) = 0.5

Explanation

Explanation

b:

\Phi(0) = 0

Explanation

Explanation

c:

\Phi(0) = 1

Explanation

Explanation

d:

\Phi(0) = -1

Explanation

Explanation

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