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ISyE 6644 — Test 3 Solutions — Fall 2008
This test is open notes, open books. You have 90 minutes. Good Luck!!
Put your answers here...
(1a) (1b) (1c)
(1d) (1e) (1f)
(1g) (1h) (1i)
(1j) (1k) (1l)
(2a) (2b) (2c)
(2d) (2e) (2f)
(2g) (2h) (2i)
(2j) (2k) (2l)
(2m) (2n) (2o)
(2p) (2q) (2r)
(3a) (3b) (3c)
1. (3 pts each) Short-answer questions on various topics.
(a) Ifappropriate constantX and Y have joint p.d.f.c, find Ef[X(x,y]. ) =
cxy2, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, for some
, 1
Solution: X and Y are clearly independent (by the multiplication
criterion). It’s then easy to see that fX(x) = 2x, 0 ≤ x ≤ 1, and
then E[X] = 2/3. ¤
(b) TRUE or FALSE? If X is a continuous random variable that’s
always positive, we have E .
Solution: TRUE. This follows since 2xP(X >
Z
x)dx = Z 2xZ Z f(y)dy dx
∞
∞ ∞
00 x
∞ y
=Z 2xf(y)dxdy
0 0
¤
(c) TRUE or FALSE? The K-S test is used to test for goodness-of-fit.
Solution: TRUE. ¤
(d) MULTIPLE CHOICE. For the probability density f(x) = x2/9, 0
≤= 3x ≤√U 3,, an inverse-transform random-variate generation
algorithm is (A) X
(B) X = U2/9, (C) X = (9U)1/3, (D) X = 3U1/3, (E) none of the
above.
ISyE 6644 — Test 3 Solutions — Fall 2008
ISyE 6644 — Test 3 Solutions — Fall 2008
This test is open notes, open books. You have 90 minutes. Good Luck!!
Put your answers here...
(1a) (1b) (1c)
(1d) (1e) (1f)
(1g) (1h) (1i)
(1j) (1k) (1l)
(2a) (2b) (2c)
(2d) (2e) (2f)
(2g) (2h) (2i)
(2j) (2k) (2l)
(2m) (2n) (2o)
(2p) (2q) (2r)
(3a) (3b) (3c)
1. (3 pts each) Short-answer questions on various topics.
(a) Ifappropriate constantX and Y have joint p.d.f.c, find Ef[X(x,y]. ) =
cxy2, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, for some
, 1
Solution: X and Y are clearly independent (by the multiplication
criterion). It’s then easy to see that fX(x) = 2x, 0 ≤ x ≤ 1, and
then E[X] = 2/3. ¤
(b) TRUE or FALSE? If X is a continuous random variable that’s
always positive, we have E .
Solution: TRUE. This follows since 2xP(X >
Z
x)dx = Z 2xZ Z f(y)dy dx
∞
∞ ∞
00 x
∞ y
=Z 2xf(y)dxdy
0 0
¤
(c) TRUE or FALSE? The K-S test is used to test for goodness-of-fit.
Solution: TRUE. ¤
(d) MULTIPLE CHOICE. For the probability density f(x) = x2/9, 0
≤= 3x ≤√U 3,, an inverse-transform random-variate generation
algorithm is (A) X
(B) X = U2/9, (C) X = (9U)1/3, (D) X = 3U1/3, (E) none of the
above.
ISyE 6644 — Test 3 Solutions — Fall 2008
