Virtual Labs

1: What percentage of the area lies between

-2\sigma

and

2\sigma

(where

\sigma^2

is the variance) for a Gaussian Distribution?

a: 90% Explanation

Explanation

b: 5% Explanation

Explanation

c: 70% Explanation

Explanation

d: 95% Explanation

Explanation

2: A random variable

X

follows a standard normal distribution and another random variable

Y

is given as

Y = 3\cdot X +2

. If we know

\mu_X=0

and

\sigma_X=1

, then what are the mean (

\mu_Y

) and standard-deviation (

\sigma_Y

) of Y?

\mu_Y=0

and

\sigma_Y=1

Explanation

\mu_Y=2

and

\sigma_Y=1

Explanation

\mu_Y=2

and

\sigma_Y=3

Explanation

\mu_Y=0

and

\sigma_Y=3

Explanation

3: Let

X \sim N(0, I_2)

be a two-dimensional standard normal random vector, where

I_2

is the

2\times2

identity matrix. What is the distribution of

Y = AX

, where

A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}

Y \sim N(\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix})

Explanation

Y \sim N(\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 5 & 2 \\ 2 & 1 \end{bmatrix})

Explanation

Y \sim N(\begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix})

Explanation

Y \sim N(\begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 5 & 2 \\ 2 & 1 \end{bmatrix})

Explanation

4: Let

X \sim N(\mu_X, \Sigma_X)

be a Gaussian random vector in

\mathbb{R}^3

, where

\mu_X = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}

and

\Sigma_X = \begin{bmatrix} 1 & 0.5 & 0 \\ 0.5 & 2 & 0.3 \\ 0 & 0.3 & 1 \end{bmatrix}

. Find the marginal distribution of

X_1

and

X_3

\begin{bmatrix} X_1 \\ X_3 \end{bmatrix} \sim N\left( \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \right)

Explanation

\begin{bmatrix} X_1 \\ X_3 \end{bmatrix} \sim N\left( \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 0 & 1 \end{bmatrix} \right)

Explanation

\begin{bmatrix} X_1 \\ X_3 \end{bmatrix} \sim N\left( \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 & 0 \end{bmatrix} \right)

Explanation

\begin{bmatrix} X_1 \\ X_3 \end{bmatrix} \sim N\left( \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \right)

Explanation

5: When does a bivariate Gaussian pdf has negative covariance?

a: When the slope of the principal axis of its pdf is positive Explanation

Explanation

b: When the slope of the principal axis of its pdf is negative Explanation

Explanation

c: When the slope of the principal axis of its pdf is

0

Explanation

d: None of the above Explanation

Explanation

Gaussian Random Vectors