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The solutions to the second midterm exam of the university of illinois, urbana-champaign, department of electrical and computer engineering, ece 313: probability with engineering applications, fall 2003. The solutions include answers to multiple-choice questions about probability distributions, including uniform, normal, exponential, and discrete distributions. The document also includes calculations for expected values and conditional probabilities. Useful for students enrolled in the ece 313 course or those preparing for exams on probability theory and engineering applications.
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Department of Electrical and Computer Engineering
ECE 313: Probability with Engineering Applications-Fall 2003
Second Midterm Exam Solution
Name Solution Key
Section: 10:00 AM 11:00 AM
Score 100
Problem Pts. Score 1 36 2 4 3 5 4 5 5 10 6 15 7 13 8 12 Total 100
Please do not turn this page over until told to do so.
You may not use any books, calculators, or notes other than two handwritten two-sided sheets of 8.5” x 11” paper.
(a) If X ∼ U [0, 1], then X^2 ∼ U [0, 1]. T/F False
(b) If X ∼ N (0, 1), then Y = 2X + 1 ∼ N (1, 4). T/F True
(c) Let X be a uniform RV with mean μ and variance σ^2 and Y be a Gaussian RV with the same mean and variance. Then P (|X − μ| > σ) = P (|Y − μ| > σ). T/F False
(d) Let X be an exponential RV with parameter λ. Then P (X > 5 |X > 2) = e−^3 λ. T/F True
(e) Let X ∼ N (0, σ^2 ) and Y = X^2. Then E(Y ) = σ^2. T/F True
(f) Let FX (u) be the CDF of a discrete RV X. If F (^) X′ (a) = 0, then P (X = a) ≡ 0. T/F True
(g) Let (X, Y ) be a discrete random vector taking on values {(ui, vj ), i, j = 0, 1 ,.. .}. Assume that the joint pmf of X and Y is pX,Y (ui, vj ) and the pmf of X is pX (ui). Then pX (ui) ≥ pX,Y (ui, vj ) for all vj. T/F True
(h) Let (X, Y ) be a discrete random vector taking on values {(ui, vj ), i, j = 0, 1 ,.. .}. Assume that the joint pmf of X and Y is pX,Y (ui, vj ) and the conditional pmf of X given that Y = vj is pX|Y (u|vj ). Then pX|Y (u|vj ) ≥ pX,Y (u, vj ) for all values of u and vj. T/F True
(i) Let (X, Y ) be a continuous random vector defined over the entire 2D plane. Assume that the joint pdf of X and Y is fX,Y (u, v) and the pdf of X is fX (u). Then fX (u) ≥ fX,Y (u, v) for all values of v. T/F False
(j) Let (X, Y ) be a continuous random vector defined over the entire 2D plane. Assume that the joint pdf of X and Y is fX,Y (u, v) and the conditional pdf of X given that Y = v 0 with fY (v 0 ) > 0 is fX|Y (u|v 0 ). Then fX|Y (u|v 0 ) ≥ fX,Y (u, v 0 ) for all values of u and v 0. T/F False
(k) Let X be a random variable characterizing the lifetime of a system with hazard rate function λ(t). Then, P (X > t) =
∫ (^) t 0 λ(τ^ )dτ^.^ T/F False
(l) Let X and Y be two random varibales related to each other by Y = g(X) with g(·) being a montonic function. Then P (Y > a) = P (X > g−^1 (a)) is always true. T/F False
fX (u) = λe−λu, u ≥ 0
FX (u) =
∫ (^) u
0
fX (t)dt = 1 − e−λu, u ≥ 0
Y = F (^) X− 1 (x) = −
λ ln(1 − x).
(15 Pts.)
P (min{X 1 , X 2 ,... , Xn} < a) = 1 − P (min{X 1 , X 2 ,... , Xn} ≥ a) = 1 − P (X 1 ≥ a, X 2 ≥ a,... , Xn ≥ a)
= 1 −
∏^ n
i=
P (Xi ≥ a)
= 1 − e−λna.
(a) P (|X| < 3) (b) P (X^2 − 3 X + 2 < 0)
Note that
FX (u) = Φ
u − 1 2
Φ(−u) = 1 − Φ(u).
(a)
P (|X| < 3) = P (− 3 < X < 3) = FX (3) − FX (−3) = Φ(1) − Φ(−2) = Φ(1) + Φ(2) − 1.
(b)
P (X^2 − 3 X + 2 < 0) = P ((X − 2)(X − 1) < 0) = P (1 < X < 2) = FX (2) − FX (1)
= Φ
