~ry
RO
iupport
nt~grity
ources
- ,,,,,.,,,,-
D Questiolil 1
1 pts ' I ' I I
I I I I
'/, I , , I ' It I
", /, 'i ! l1W~ ils a ~~,~dorn iflarl~ble with c.d.f. function F(x). Suppose X has three possible values 1,2, or 3. 11' 11'i'II~ I I 111/11/!w~~t 1$, 1~r p.ropjibillties of X being 1, 2, or 3 are all positive, and , I' i/11 11' 111 1/PitX = · l} + l' (X = 2) + P (X = 3) = 1, where P (X = i) is the probability of X ' \11
1
1/1
1,, '
1
1
l\1\ ~q4als to L'( .I
\ I l/1 w, 111
I Ii II ,ii ' I I 1, .11
' I 'j
'11, 11;, I
1
,11, Please select all co
ect options: ' I
,,
I
1 tr
LI F(3.7) -F~2.2) equii!ls to the probability of X equals to 3. ---------·---- ......... - · XXXXXXXXXXO F(l.5) is strictly greater than F(l.1) .
J F(z) has three upper jumps
-··-··-·-··••·----·-··-· XXXXXXXXXXra ~ is a discrete random variable.
J F'(2) doesn't exist.
O There is some z > 0 such that F'(z) > 0 .
D \ Question 2
\ Let X be a non-cgnstant random variable with c.d.f F(x). Please select all co
ect options:
Jmft1 '!ii'
1 pts
D
I•. I: rl ' I l, I
1 'n 1 · 1, '
Let X ,be a 111~ra-comstant random vartal'.Jle with c.d.f F(;,; ). Please, select all co
ect options:
0 If X has a standard normal distribution, F( m) < 1 for all real number x.
Ii If X is a continuo.us random variable, the slope of it's cdf is always (weakly) steeper than the slope of cdf
of2X.
Ii If X has a standard normal distribution, then F'(O) > F'(m) for all x other than 0.
l!'J If X has a standard normal distribution, F( x) = 0 for some real number x.
D Y is another random variable. If X and Y have the identical cdf function, X "' Y. (X "' Y means the value of X
and Y are equal under every state.)
l!'J Since X is a random variable, -Xis a random variable. It is possible that X and -X are independent
Question 3 1 pts
Bob is flipping a coin. He will continue unless head shows up two times in a roll. Suppose
probabilities of the coin lands on head or tail are even. What is the probability of he stops exactly
at the fourth flipping? (Hint: carefully consider the cases in which he ends before the fourth flip)
I 0.25
... hao123 ~ g -. M Gmail â– V,
1 1.
1 1 1 rtction, X"' Y. (X "' means the value of X 'I ' I I
a~·",, XXXXXXXXXXll\l\\\1i\,111,lll
D Question3 1 pts
Bob is flipping a coin. He will continue unless head shows up two times in a roll. Suppose
probabilities of the coin lands on head or tail are even. What is the probability of he stops exactly
at the fourth flipping? (Hint: carefully consider the cases in which he ends before the fourth flip)
\ 0.25
D Question4 1 pts
Let the state space n = {1, 2, 3, 4, 5, 6}. Each state happens with same probability. Let
X, Y, Z be three random variables.
X = 1 if the state is an odd number(1, 3, or 5), X = 0 otherwise.
Y = 1 if the state is an even number(2, 4, or 6), Y = 0 otherwise.
Z = 1 if the state is strict! smaller than 5, Z = 0 otherwise.
CK 5 ' o' a • •• " l•
l'lll3i B n muff lfd 1h 'I ~ lâ– Bi1lU. ~ ~ -U,
I
Let the state space {l = {1, 2, 3, 4, 5, 6}. Each state happens with same probability. Let
X, Y, Z be three random variables.
X = 1 if the state Is an odd number(1, 3, or 5), X = 0 otherwise.
Y = 1 if the state is an even number(2, 4, or 6), Y = 0 otherwise.
Z = 1 If the state is strictly smaller than 5, Z = 0 otherwise.
Recall that two random variables are independent if knowing the value of one of them does not
change the probabilities for the other one.
Please select all co
ect options:
El X and Z are independent.
El The variances of X and Y are equal.
El X + Y has larger variance than X
O X and Y are ind~porident.
O X and Y are negatively co
elated
,.ucsd.edu/courses/25144/quizzes/6420O/take Ill , ,111J'jl\lil
· , ,. , 111 'JI I 11111 , 1
' ,I I ii ' ', I ' ' I ' ' ,, '
IJ ••• _.. l t I A•t- 1••• ..,
~3i ml ffl~ m m~~ afD hao12~ Th5,4 ,liill;tib ' XXXXXXXXXXDfffir~ !11 '. i5?1'1t 1.mfillilH1fl> I , I ~ ~ ~ F'J ~ M:, Gm.ail D ·voulube 1~ 11 I '1" '·
' t; ,1
i◄ • I I
j. ,,
I if
! I "
11
l : ' I
I I
• j H
J I
D
l I I
Question 5 ·
1
1
1 l ',111
I I ' ' 11 I, I
, '1 Ii \;•,, I 1:I\ I, il1'1 1111 I
I·
1 pts
I I I ' ii I I
Let the st~te space n { a1, ~ I, ~3 ~ a4}. Three states happen with same probability (0.25).
Let X Y be two randolJl variables., 1
, I I Iii I '
X = 4 regardless of the state. Y = 16 if state is a1 , Y = 0 otherwise.
I '
Please select all co
ect options:
O The expectation of X is equal to the expectation of Y
-----
□ Consider random varia~ les -v'X and ✓Y, we have E( ✓]{) > E( ✓Y).
!J E{Y) = 4
!J Consider rando~ variables X 2 and Y 2 , we have E(X
2
) < E(Y
2
)·
1 pts
I
D LQ~u~es~n:·o:n~6:· ..,L• __ _L: _J~~--.!-' XXXXXXXXXX
Let X, Y, Z be the three assets. Suppose C (X Y) - 1 Cov(X, Z) = - 2, Cov(Y, L
V AR(X) = 1, V AR(Y) = 2, V AR(Z) = 5. ov ' - ' .
~ - --
[jj C!l (f
~------·-0CH .,S II) '; .... i-· 0 lit ,c ,, ·"''
Let X, Y, Z be the three assets. Suppose
V AR(X) = l, V AR(Y) = 2, V AR(Z) = 5.Cav(X, Y) = l,Cau(X,Z) = -2,Cau(Y,1
Let the portfolio be ( Wx, Wy, Wz) , which portfolios yield the minimal variance? (Hrnt can we
find a combination of X and Y that is perfectly negatively co
elated with Z?)
â—„
O (wx, wy, wz) = (0.8, 0, 0.2)
@ (wx,wy,wz) = (½, ¾, ½)
0 (wx, wy,wz) = (½ , ½, ½)
0 (wx,wy,wz) = (~. ~' t)
â–º
D I Question 7 1 pts
D Question 7
In this question, all assets have positive variances.
Please select all co
ect options:
l Given N assets, if there is no pair of assets that is perfectly negatively co
elated, then the smallest
variance from portfolio must be positive.
O If X and Y are two negatively co
elated assets, there always exists a portfolio with no variance.
O If two assets are not perfectly positively co
elated, any combination yields a smaller variance than each
single asset.
D Given N asset, if there are two negatively co
elated assets, the optimal portfolio involves at least two
assets.
J If two assets arc not perfectly positively co
elated and have same variance, any combination yields a
smaller variance than each single asset.
1 pts
et X and Y be two asset~. ;Y;JiR(X) ' .4 and V AR(Y) = 9. However, their co
elation Is
nknown. What is the
1
l~rgeM possible value for V AR(2X + Y)?
!
17
D Question 9
I I
1 pts
Let X and Y be two assets, V AR(X) = l, V AR(Y) = 4, GOV(X, Y) = 1.5 . What is the
portfolio weight on X that gives the smallest variance?
125
D Question 10 1 pts
Let X , Y, Z be the t hree assets, with V AR(X) = V AR(Y) = V AR(Z ) = l. Z is
independent from X and Y , and OOV(X, Y) = 0.2 . What is the portfolio weight on Z that
gives the smallest variance? (Hint- since Z is independent with X and Y you may express w
D
,I I ' 11! 1111 hl'I
I
, II.et~ arid Y be two assets,, V AR(X) = l i V AR(Y) = 4, COV(X, Y) = 1.5. What is the
portfolio weight on X that give$ the smallest variance?
1 pts
Let X, Y, Z be the three assets, with V AR(X) = V AR(Y) = V AR(Z)