## 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 $$P(A \cup (B \cap C))$$.