Virtual Labs

The median of a distribution is the value of the random variable that separates the higher and lower halves of a distribution. In Standard normal distribution, the value of median is ___________?

a: 2 Explanation

Explanation

b: 1 Explanation

Explanation

c: 0 Explanation

Explanation

d: Not Fixed Explanation

Explanation

Let

X \sim N(2, 4)

and

Y \sim N(3, 9)

be two independent Gaussian random variables. What is the distribution of

Z = X + Y

Z \sim N(5, 13)

Explanation

Z \sim N(5, 16)

Explanation

Z \sim N(5, 81)

Explanation

Z \sim N(5, 25)

Explanation

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

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

Explanation

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

Explanation

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

Explanation

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

Explanation

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)

\rho(X_1, X_2) = 0.7

Explanation

\rho(X_1, X_2) = 1.0

Explanation

\rho(X_1, X_2) = 0.49

Explanation

\rho(X_1, X_2) = 0.5

Explanation

A random variable

X

follows a standard normal distribution

N(0, 1)

. What is

P(-1 \leq X \leq 1)

? (Refer to the CDF table for standard normal distribution)

0.68

Explanation

0.95

Explanation

0.50

Explanation

0.99

Explanation

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)

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

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

Explanation

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

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

Explanation

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

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

Explanation

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

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

Explanation

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

x = 0

\Phi(0) = 0.5

Explanation

\Phi(0) = 0

Explanation

\Phi(0) = 1

Explanation

\Phi(0) = -1

Explanation

Gaussian Random Vectors