3770 1
ISyE Practice Test #2 + Solutions
Fall 2007
1. What does the d in c.d.f. mean? Solution: distribution 2. Suppose X has p.d.f. f (x) = cx, 0 x 3. Find c. Solution:
2 9 3 0
1=
cx dx
3. Suppose X has p.d.f. f (x) = 4x3 , 0 x 1. Answer all of the following questions. (i) Find Pr(X 1/2). Solution:
0 1/2
4x3 dx = x4 |0
1/2
=
1 = 0.625 16
(ii) Find Pr(X 1/2 | X 1/4). Solution: x4 |1/4 x 4 |1 1/4 (iii) Find E[X]. Solution:
0 1 1/2
=
1 = 0.0588 17
4x4 dx =
4 = 0.8 5
(iv) Find E[X 6 ]. Solution:
0 1
4x9 dx =
2 = 0.4 5
4. Suppose that X is a discrete random variable with p.m.f. Pr(X = 1) = 0.5, Pr(X = 0) = 0.2, and Pr(X = 2) = 0.3. Find E[X], Var(X), and E[X 3 ].
2 Solution: By denition of expectation and variance for discrete random variables: E[X] = = Var[X] = = E[X 3 ] = = Pr(X = 1) (1) + Pr(X = 0) 0 + Pr(X = 2) 2 0.5 + 0 + 0.6 = 0.1 E[X 2 ] E[X]2 0.5 (1)2 + 0.2 (0)2 + 0.3 (2)2 (0.1)2 = 1.69 Pr(X = 1) (1)3 + Pr(X = 0) (0)3 + Pr(X = 2) (2)3 0.5 + 0 + 2.4 = 1.9.
2
5. Multiple choice. The theorem that states that Pr(|X E[X]| < ) Var(X)/ known as (a) The Law of the Unconscious Statistician. (b) Chebychevs Inequality. (c) Bayes Theorem. (d) The Central Limit Theorem. Solution: b 6. Suppose E[X] = 3 and Var(X) = 2. Find E[3X 1]. Solution: 3 E[X] 1 = 9 1 = 8 7. Suppose E[X] 3 = and Var(X) = 2. Find Var(3X 1). Solution: 9 Var(X) = 18
is
8. Suppose X has the Bernoulli distribution with parameter p. What is the probability that X = 0? Solution: 1 p 9. The probability that a Britney Spears CD will sell a million copies is 0.7. Consider 5 dierent Britney CDs. Whats the probability that exactly 2 of these 5 will sell over a million copies? Solution: 5 (0.7)2 (0.3)3 = 0.1323 2
3 10. What is the mean of the exponential distribution with parameter = 0.1? Solution: 10 1 1 = 0.1
11. Suppose that X has p.d.f. f (x) = 2x, 0 < x < 1. Find the p.d.f. of Y = n(X). Solution: 2e2y , < y < 0 or 0 < ey < 1 G(y) = Pr(ln x y) = Pr(x ey ) =
ey 0
2x dx
y
= x 2 |e 0
= e2y , 0 < ey < 1 or < y < 0 12. What does the m in m.g.f. stand for? Solution: moment (moment generating function)
13. If X is a random variable with m.g.f. 0.4et + 0.6, nd Var(X). Solution: 0.24 If Mx (t) = pet + q X Bern(p) Var(X) = pq Thus, X Bern(0.4) in this example Var(X) = 0.24 14. True or False? If X is a discrete random variable with m.g.f. MX (t), then E[X] = d MX (t). dt Solution: False d E(X) = Mx (t)|t=0 dt 15. TRUE or FALSE? The Law of the Unconscious Statistician states that, if g() is a continuous function and X is a random variable, then E[g(X)] = g(E[X]). Solution: False