University Final Exam ECE 313
of Illinois Page 1 of 3 Fall 1999
1. Check the appropriate box for each of the statements below. Answers need not be justified.
However, in order to discourage guessing, you will be penalized for wrong answers.
(a) (+4 points for a correct answer, –4 points for a wrong answer, and 0 points for no answer)
Which of the following statements are true for all events A and B such that 0 < P(A) < 1 and
0 < P(B) < 1?
TRUE FALSE
■ ■ P(A|B) > P(A)
■ ■ P(A|B) + P(A|Bc) = 1
■ ■ P(A|B) + P(Ac|Bc) = 1
■ ■ P(A|B) + P(Ac|B) = 1
■ ■ If P(A) = P(B), then P(A|B) = P(B|A)
■ ■ If P(A|B) = P(B|A), then P(A) = P(B)
■ ■ If P(A|B) = P(B|A), then A and B are independent
■ ■ P(A ∩ B) ≤ P(A)P(B) with equality if and only if A and B are independent
■ ■ P(B|A)P(A) + P(A|Bc)P(Bc) = P(A)
■ ■ P(A|B) = P(B|A)P(B)/P(A)
(b) (+8 points for a correct answer, –2 points for a wrong answer, and 0 points for no answer)
Which of the following four statements are true for all events A and B such that
0 < P(A) < 1, 0 < P(B) < 1?
①P(A ∩ B) ≤ min{P(A), P(B)} ➁P(A ∪ B) ≥ [P(A) + P(B)]/2
➂P(A ∩ B) ≥ P(A) + P(B) – 1. ➃P(A ∩ B) ≤ P(A|B)
■Only ① and ➂ are true statements.
■Only ➁ and ➂ are true statements.
■Only ①, ➂, and ➃ are true statements.
■All four are true statements.
■None of the above: only the following are true statements
(c) (+8 points for a correct answer, –2 points for a wrong answer, and 0 points for no answer)
Which of the following four statements are NOT properties of all CDFs?
① P{X > b} = 1 – FX(b). ➁If FX(a) < FX(b), then a < b.
➂If a < b, then FX(a) < FX(b). ➃FX(u) = 1/2 for some u, –∞ < u < ∞.
■Only ➁ is not a property of all CDFs.
■Only ➁ and ➂ are not properties of all CDFs.
■Only ➂ is not a property of all CDFs.
■Only ➂ and ➃ are not properties of all CDFs.
■You blew it, Professor! All four are properties of all CDFs.
