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Aim

Theory

Pre-test

Procedure

Experiments

Post-test

References

Contributors

Feedback

Functions of a Random Variable

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Beginner

Intermediate

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1. Let

X

be a continuous random variable with PDF:

f_X(x) = \begin{cases} x^2\left(2x+\frac{3}{2}\right) & 0 < x \leq 1 \\ 0 & \text{otherwise} \end{cases}

Find the variance

Var(X)

.

a:

\frac{21}{36}

Explanation

Explanation

b:

\frac{61}{36}

Explanation

Explanation

c:

\frac{51}{36}

Explanation

Explanation

d:

\frac{71}{36}

Explanation

Explanation

2. Let

A = XY

denote the area and

L = 2(X + Y)

the length of the perimeter of a rectangle with length

X

and height

Y

, such that

X

and

Y

are independent, and uniformly distributed on the interval

[0, 1]

. Find

\mathbb{E}[A]

a: 0.25 Explanation

Explanation

b: 0.5 Explanation

Explanation

c: 0.8 Explanation

Explanation

d: 0.4 Explanation

Explanation

3. Let X be the outcome of a single roll of a fair six-sided die. What is the expected value of the random variable

Y = 2X + 1

?

a: 3.5 Explanation

Explanation

b: 7 Explanation

Explanation

c: 8 Explanation

Explanation

d: 4.5 Explanation

Explanation

4. Let

X

be a continuous random variable uniformly distributed on the interval

[0, 1]

. Find the expected value,

\mathbb{E}[Y]

, for the random variable

Y = X^2

.

a: 1/2 Explanation

Explanation

b: 1/4 Explanation

Explanation

c: 1/3 Explanation

Explanation

d: 1 Explanation

Explanation

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