Please see the **detailed homework policy document** for information about homework formatting, submission, and grading.

## Exercise 1

**(a)** Evaluate the following integral. Do **not** use a calculator or computer, except to check your work.

\[
\int_{0}^{\infty}x e^{-2x}dx
\]

**(b)** Evaluate the following integral. Do **not** use a calculator or computer, except to check your work.

\[
\int_{0}^{\infty}x e^{-x^2}dx
\]

## Exercise 2

Find the value \(c\) such that

\[
\iint\limits_A cx^2y^3 dydx = 1
\]

where \(A = \{ (x,y) : 0 < x < 1, \ 0 < y < \sqrt{x} \}\). Do **not** use a calculator or computer, except to check your work.

## Exercise 3

Suppose \(S = \{2, 3, 4, 5, \ldots \}\) and

\[
P(k) = c \cdot \frac{2^k}{k!}, \quad k = 2, 3, 4, 5, \ldots
\]

Find the value of \(c\) that makes this a valid probability distribution.

## Exercise 4

Suppose \(S = \{2, 3, 4, 5, \ldots \}\) and

\[
P(k) = \frac{6}{3^k}, \quad k = 2, 3, 4, 5, \ldots
\]

Find \(P(\text{outcome is greater than 3})\).

## Exercise 5

Suppose \(P(A) = 0.4\), \(P(B^\prime) = 0.3\), and \(P(A \cap B^\prime) = 0.1\).

**(a)** Find \(P(A \cup B)\).

**(b)** Find \(P(B^\prime \mid A)\).

**(c)** Find \(P(B \mid A^\prime)\).

## Exercise 6

Suppose:

- \(P(A) = 0.6\)
- \(P(B) = 0.5\)
- \(P(C) = 0.4\)
- \(P(A \cap B) = 0.3\)
- \(P(A \cap C) = 0.2\)
- \(P(B \cap C) = 0.2\)
- \(P(A \cap B \cap C) = 0.1\)

**(a)** Find \(P((A \cup B) \cap C^\prime)\).

**(b)** Find \(P(A \cup (B \cap C))\).