Virtual Labs

1. A random variable

X

is uniformly distributed in the interval (0,3). Another random variable

Y = e^{-2X}

. The PDF of

Y

\frac{ln Y}{6} \text{, } e^{-6} \leq Y \leq 1

Explanation

\frac{ln Y}{8} \text{, }e^{-2} \leq Y \leq 1

Explanation

\frac{1}{6Y} \text{, }e^{-6} \leq Y \leq 1

Explanation

-2 e^{-2Y} \text{, } 0 \leq Y \leq 3

Explanation

2. A random variable

X

is uniformly distributed over

[-1,4]

. A new random variable

Y

is defined such that

Y = 2X+5

. Then the probability density function

f_Y(y)

is given by:

f_Y(y) = \left\{ \begin{array}{ll} \frac{1}{6}, & 1 \le y \le 7 \\ 0, & \text{Otherwise} \end{array} \right.

Explanation

f_Y(y) = \left\{ \begin{array}{ll} \frac{1}{10}, & 3 \le y \le 13 \\ 0, & \text{Otherwise} \end{array} \right.

Explanation

f_Y(y) = \left\{ \begin{array}{ll} \frac{1}{8}, & 3 \le y \le 13 \\ 0, & \text{Otherwise} \end{array} \right.

Explanation

f_Y(y) = \left\{ \begin{array}{ll} \frac{1}{8}, & 3 \le y \le 11 \\ 0, & \text{Otherwise} \end{array} \right.

Explanation

3. Let

X

and

Y

be two statistically independent random variables uniformly distributed in the ranges

(-1, 1)

and

(-2, 1)

respectively. Let

Z = X + Y

. Then the probability that

Z \leq -2

a: Between 0.8 to 0.9 Explanation

Explanation

b: Between 0.4 to 0.6 Explanation

Explanation

c: Between 0.04 to 0.06 Explanation

Explanation

d: Between 0.08 to 0.1 Explanation

Explanation

4. Let

X

and

Y

be two independent random variables where

X

is exponentially distributed of rate

\lambda_1

and

Y

is exponentially distributed of rate

\lambda_2

. Let

Z = \min(X, Y)

. Find the density of

Z

given by

f_Z(a)

a: 0 Explanation

Explanation

f_Z(a) = \left\{ \begin{array}{ll} (\lambda_1 - \lambda_2)e^{-(\lambda_1 + \lambda_2)a} & \text{for } a \geq 0 \\ 0 & \text{otherwise} \end{array} \right.

Explanation

f_Z(a) = \left\{ \begin{array}{ll} (\lambda_1 + \lambda_2)e^{-(\lambda_1 + \lambda_2)a} & \text{for } a \geq 0 \\ 0 & \text{otherwise} \end{array} \right.

Explanation

f_Z(a) = \left\{ \begin{array}{ll} (\lambda_1 + \lambda_2)e^{-(\lambda_1 - \lambda_2)a} & \text{for } a \geq 0 \\ 0 & \text{otherwise} \end{array} \right.

Explanation

5. Let

X

be a discrete random variable with range

R_X = \left\{ 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi \right\}

. Find

\mathbb{E}[\sin(X)]

\frac{\sqrt{2}}{5}

Explanation

\frac{2\sqrt{2}}{5}

Explanation

\frac{1 + \sqrt{2}}{5}

Explanation

\frac{2 + \sqrt{2}}{5}

Explanation

Functions of a Random Variable