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Unformatted text preview: Page I ofIO Econ 15B: Probability and Statistics Midterm, May 10, 2007
Kent Johnson, S n 2007 Test Form: B Choose the best answer to each question. Caution: Some questions may strike you as similar to
questions you have seen elsewhere. Be sure to consider the questions and the answers carefully;
some questions] answers may have been changed. Scratch paper is provided at the end of this exam. 1. Suppose f is the pdf of a uniform distribution on the interval (3, 5). Then the derivative of f at the value 4.5 is:
a. 0 b. 1
@ .5
d 00
6. None of the above 2. If X is a random variable, then E(X) Is a random variable, too.
.. Is a function, but not a random variable.
c. Is a number.
(1. Is a distribution.
e. None of the above. 0' 3. The Central Limit Theorem tells us that as n ——> 00, 1;: [X‘ _ y] is distributed as N(O, 1).
n i=1
This means that the following are also distributed as N(0, 1): e. None of the above Page 3 of“) xii/,6
9. Suppose X ~ N(0, 1), and Y ~ N(u, 0'2). Then: 3‘ X33” '0 7T2 ” "
® X=:___1” /
0' 2
c. X=[Y—2’u]
0' 2
d_ X = 0' 6. None of the above. 10. If F is the cdf of a continuous distribution, then F (x) Q Is always a value between 0 and 1
. Is always greater than the pdf of the distribution 7 c. Is always less than the pdf of the distribution
(1. Could be any nonnegative number at all.
0 6. None of the above. 11. Let {X1,...,Xn} be a random sample of the continuous random variable X. Then E(X#) = :(X; ‘1”) 0 @ Is true.
{3 .r Doesn’t make sense, since they’re not both numbers.
c. May be true sometimes and false other times, it all depends on X. d. Istrueonlyasn—>oo.
e. None of the above; it all depends on which distribution is at issue. 12. The probability of a continuous random variable yielding a value more than 4 stande
deviations away from its mean is certain to be less than: [Hint use Chebyshev’s inequality] .4
. I
1:51;: WW >ti <7;
/ d. 1/.4 45/ __
@ Noneoftheabovelervﬂq)> fsx f; :— /® 13. Let {x1,. . .,xn} be a given data set that resulted ﬁ‘om randomly sampling from a particular
discrete random variable X. Then E(X) = f a. Is always true.
. Is always false, because a? is a number, but E(X) is not. c. Is always false, but for some other reason.
@ Is true for some data sets, and false for others.
e. None of the above. Page 5 ofIO 18. You are attempting to discover the average personal income in Orange County, and you
sample a variety of female Orange County residents. Then your sample:
a. Is unproblematic.
b. Is problematic because it violates the requirement of independence.
c. Is problematic because it violates the requirement of identical distribution.
d. Both b and 0.
® None of the above. 19. Let f be a pdf of a continuous distribution. As discussed in class and in the book, f is naturally
represented as: An integral
b. A derivative
0. A probability function
(1. b and c.
e. a and c. 20. The skew of the normal distribution can be altered by:
a. Changing the value of the mean only. Changing the value of the variance only.
Q23 Changing the value of the mean and the variance.
I Changing either the value of the mean or of the variance, but not both.
$( @The skew cannot be altered. 21. If f ' the pdf of an unknown continuous distribution, then ﬁx)
Is always a value between 0 and 1.
bf Is always a value between —1 and 1. OFF 5 1 7 I? 0
c. Could be any number at all. 07 ' O
d Could be any nonnegative number.
4; e None of the above. r—Hfm— 22. Suppose you examine the distribution of a random sample of averages {f1,..., 37m } , where each 27,. = 2 y, is itself a random sample. The Central Limit Theorem tells us that this
i=l distribution will come to resemble a normal distribution as:
a. m increases.
@ 11 increases.
0. In and n both increase.
(1. Either m or 11 increase.
e. None of the above. Page 7 of“) Extra Credit. 28. Under what conditions does E(X2) = 02?
a. Never.
Always. Only when X is discrete, not continuous (but not necessarily for all discrete
variables). d. Only when X is continuous, not discrete (but not necessarily for all continuous
variables). e. None of the above 29. Consider the following attempt to express f, the pdf of N01, 0'2), in terms of :1), the pdf of
N(0, 1): foo: 1 exp[_(x—p)2]_1 1 exp[_x2—2xp+p2)= 2 42m: 20‘2 — ; J27r 20'
1 2 3
1 1 x2 Jewp: 1 xii—p2 l x2
— ex — + = —ex ex — =
0' «127: p[ 202 0'2 0' p 02 1127: p 202
4 5
l xp—pz 1 x2 ‘0 1 xp— #2 _az
—ex ex —— = —~ex
0' p[ 0'2 J «I 271' p[ 2 0' p 0'2 ﬁx)
6 7 a. The proof is correct.
b. There is an error in step 3 to 4 only
c. There is an error in step 4 to 5 only. © There is an error in step 5 to 6 only. e. There is more than one error in the proof. END OF EXAM P “5°?”ng 11.
12.
13.
14. 15. {HQ—’— Econometrics — Homework #1 C] 2 2 B] = 0 61 = 2
62 = 3 .03 .26 .01
= 0 .01 .38 .01
e; = 3 .01 .26 .01 Present the marginal densities of el and e; in both tabular and graphical form
Find the means and variances of e, and eg. Show your setup. Which random variable is most
likely to take on a value different ﬁ'om its mean? Which random variable has the largest
variance? 7
Find the covariance and correlation coefficient of e] and 94. Show your setup. Are el and ea
more likely to move together, or more likely to move in opposite directions?
Find the conditional density of e1 given e; = 3. Present this density in both a table and a
graph. Find the conditional mean and the conditional variance of e1 given e; = 3.
Repeat #4 for the conditional density of e, given e; = 0. '
Repeat #4 for the conditional density of e1 given e; = 3.
Repeat #4 for the conditional density of e; given e; = 2
Repeat #4 for the conditional density of e; given at = 0
Repeat #4 for the conditional density of a; given e1 = 2
Consider the supply of labor relationship: yt = ﬁn + 6t
where y, is the wage rate, x. is the quantity of labor being supplied, at is a random disturbance,
and B is the slope of the supply curve. Graph the nine possible samples which are presented in the table presented below
Y1 = 1 3’1 = 3 Y1 = 5
y; = 2.5 .02 .16 .02
y; = 4 .06 .48 .06
y; = 5.5 .02 .16 .02 Let x1 = 12, x2 = 16. Find b" = (x1 y; + x2 ya) I (x,2 + x22 ) for each of the nine possible samples ' Find b" = max(yl/x1 , yzlxz ) for each of the nine possible samples Table and graph the probability densities of b* and b**. Find the means and variances of h" and b‘f“. Which estimator is unbiased? How can you tell? ’ ' Consider the “demand as a function of price and income” relationship Yr: thl +B2xt2+st where y, is demand, where x“ is price, and where xa is income. You have the following
information on demand, price and income demand = 3 price = 2 income = 3
2 3 1
3 1 2
1 2 2
3 2 3 Find b*, c(b*), and 90% conﬁdence intervals for [31 and [32 by hand. Now ﬁnd b*, can"), and
90% conﬁdence intervals for B] and it; using Eviews y—v. ECM (2m
4/ 24/07 Page 6 of10 23. If you have a random sample of 100 observations, and the variance of this sample is 4, what
is the standard deviation of your sample average statistic I? ? l +
5 r. a. .2 ® 2 c. 4 d. .04 e. None of the above. 24. Using the notation from our slides in class, in the situation of stratiﬁed random sampling, we have: f=l[iyi+iz,], J7=iiyi , and E=iizi .Inthis case:
i=1 i=1 n n 1i=i n2i=1
a r=l()7+§ n .— _ _ "' HI = ' ‘Z £2)=ZZ' b f=l[l+i] \MT) 23“ (u " n n1 n2
c f=—P—+—§— n1 "2 e. None of the above. 25. [Hint This is the “student requested” exam question] When X and Y are any random
variables, p2(X+Y) = ,uz(X) +,u2(Y)
a. Is always true.
b. Is never true.
0. Is true if X and Y are identically distributed.
(1. Is true if X and Y are independent.
X and Y must be both independent and identically distributed for this to be true. 26. Let X ~ N(0, 1). Then pr({0 S X S 0}) =
a. Approximately .4. b. 1 .
c. Approximately onehalf.
r / p" 6. None of the above. 27. If a question on this exam was so difﬁcult that every student simply guessed at the answer
(and no student copied another’s answer), the number of students who get the question light
would most likely come from sis—mm. mom # M‘ {pkde b. A Poisson distribution. ' , * / I r
 A uniform distribution. (5 at! {/7 f0 acm a d. A normal distribution. e. None of the above. Page 4 ofIO 14. If f is the pdf of X, where X ~ N(.L, 0'2), then it is certain that 1300 = 99%
u c.c:2 d fut)
e. None of the above 15. The variance of a random variable distributed as N(p., 62) can be expressed as:
M [_(x#)2 None of the above 16. Let F be the cdf of a continuous distribution for a random variable X. pr(a S X S b) =
a. F(b) b. F(a)
c. F(a)—F(b) @ @ F(b)—F(a) 6. None of the above 17. When A7 = lZXI. is 1the mean ofa random sample, the mean of .7, Lean? , is equal to:
7' i—l a. 71sz
b. max
\ 1
g ‘1”):
n
d. Cannot be deteimined from the information given.
@) None of the above Page 2 of“) 4. With respect to a population represented by the random variable X, the advantage of
stratiﬁed random sampling over simple random sampling is that: ' a. Stratiﬁed random sampling reduces ﬁx
b. Stratiﬁed random sampling reduces p} c. Stratiﬁed random sampling reduces 0'} @ Stratiﬁed random sampling reduces a},
6. None of the above. 5. Let be a continuous random variable whose cdf is F. Then F(x) is: Uzi/"Ii
. “None of the above 7. Suppose fis a pdf. Then jf(x)dx = 
0 a. F(0)
b. .5
c. aandb Q 1—F(0) e. None of the above. 8. Suppose ﬂy is the pdf of X, where Kheblfé, 5) and fy is the pdf of Y, where Y ~ N(3, 7). Then:
8 fx(3) =fr(3)
@ 13(3) 99(3)
c. M3) >196)
(1. Cannot be detennined from the information given.
e. None of the above. ...
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