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(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.
\[ \int_{0}^{\infty}x e^{-x^2}dx \]
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.
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.
Suppose \(S = \{2, 3, 4, 5, \ldots \}\) and
\[ P(k) = \frac{6}{3^k}, \quad k = 2, 3, 4, 5, \ldots \]
Find \(P(\text{outcome is greater than 3})\).
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 \(P(B \mid A^\prime)\).
Suppose:
(a) Find \(P((A \cup B) \cap C^\prime)\).
(b) Find \(P(A \cup (B \cap C))\).