l*******l
发帖数: 248 | 1 1.Find the number a which maximizes E(X+aY), where X and Y are random
variables.
2. You have two fair dice, A and B. You first throw A, then B, then A again,
etc. You stop the first time you see six. What is the probability that the
die A will give you the first six?
3. Suppose you have a portfolio of 100 bonds, each with a 1% yearly default
rate. What are the odds that no bonds in the portfolio default in a given
year? Estimate this to the first digit. What happens if there is
correlation between the bond defaults?
4.Suppose you have returns distributed with a 5% expected return and 20%
yearly standard deviation. What is the 1sd drawdown over a 2-year period (
in other words, what is the return for -1 standard deviation)? |
l*******l
发帖数: 248 | 2 5.Let X be a random variable. What is bigger, E(e^X) < e^(E(X))? Why?
6.Suppose you have a polygon with n vertices. In how many ways can one
triangulate this polygon? (Points for thinking of recursion right away,
points for setting it up in such a way that there is no overcount, many
points for deriving this recursion, extra points for solving it. The latter
is hard, but can be done using "generating functions". Knowing what is
Catalan's number helps) |
x********o
发帖数: 519 | 3 for the first one, you really mean variance? right? |
l*******l
发帖数: 248 | |
n******m
发帖数: 169 | |
v*******y
发帖数: 1586 | 6 第二题不难吧
等比数列？
a0=1/6, r=(5/6)^2 |
l*******l
发帖数: 248 | 7 你是说第六题？
【在 v*******y 的大作中提到】
: 第二题不难吧
: 等比数列？
: a0=1/6, r=(5/6)^2
|
n******m
发帖数: 169 | 8 not sure.
1. cov(x,y)/var(y)
2. 6/11
3. (99%)^100, estimate 37.5%, calculator says 36.6%.
if corelated, can be max 99% min 0%
4.....
5. Set E(x)=0. Suppose pdf exist. need to prove int(e^xf(x)dx))<=1. Let F(t)
=int(e^(tx)f(x)dx), then F(0)=1, F'(t)<0, So F(1)<=1.
6. Fix one vertex, consider clockwisely the first ray going out from this
vertex, it divids the polygon into 2 part. Then triangulate the 2 parts. To
gaurantee on overcount, the first part should not have any ray coming out
from the pre-fixed vertex.
Count is F(2)=1, F(3)=1, F(n)= sum_k (F(n-k+1)F(k)) where 2<=k<=n-1 |
l*******l
发帖数: 248 | 9 1.why?
3.how did you estimate? the diff seems v big lol
4. what is that u don't understand, i think it is readable.
t)
To
【在 n******m 的大作中提到】
: not sure.
: 1. cov(x,y)/var(y)
: 2. 6/11
: 3. (99%)^100, estimate 37.5%, calculator says 36.6%.
: if corelated, can be max 99% min 0%
: 4.....
: 5. Set E(x)=0. Suppose pdf exist. need to prove int(e^xf(x)dx))<=1. Let F(t)
: =int(e^(tx)f(x)dx), then F(0)=1, F'(t)<0, So F(1)<=1.
: 6. Fix one vertex, consider clockwisely the first ray going out from this
: vertex, it divids the polygon into 2 part. Then triangulate the 2 parts. To
|
l*******l
发帖数: 248 | 10 F'(t)怎么求？ 罗比大？
t)
【在 n******m 的大作中提到】
: not sure.
: 1. cov(x,y)/var(y)
: 2. 6/11
: 3. (99%)^100, estimate 37.5%, calculator says 36.6%.
: if corelated, can be max 99% min 0%
: 4.....
: 5. Set E(x)=0. Suppose pdf exist. need to prove int(e^xf(x)dx))<=1. Let F(t)
: =int(e^(tx)f(x)dx), then F(0)=1, F'(t)<0, So F(1)<=1.
: 6. Fix one vertex, consider clockwisely the first ray going out from this
: vertex, it divids the polygon into 2 part. Then triangulate the 2 parts. To
|
|
|
n******m
发帖数: 169 | 11 你是对 t 求导啊，x都看作常量.
【在 l*******l 的大作中提到】
: F'(t)怎么求？ 罗比大？
:
: t)
|
l*******l
发帖数: 248 | 12 F'(t)=int(xe^(tx)f(x)dx)怎么看出小于零？？
【在 n******m 的大作中提到】
: 你是对 t 求导啊，x都看作常量.
|
n******m
发帖数: 169 | 13 1. Forget about it. I did min Var, may not be what you are looking for.
3. (1-1%)^100 binomial formula, first few terms are approximately 1-1+1/2-1/
6+1/24
【在 l*******l 的大作中提到】
: 1.why?
: 3.how did you estimate? the diff seems v big lol
: 4. what is that u don't understand, i think it is readable.
:
: t)
: To
|
n******m
发帖数: 169 | 14 分正负两段积分。
负的那段，e^ 小于1，绝对值比 int (xfxdx) 小
正的那段，绝对值比 int(xfxdx) 大
而 int(xfxdx) 正负两段的总合是0
这个有点出乎意料之外，因为原不等式成立的原因是 e^x 是凸函数，但是这里我们没
有算二阶导。不过考虑到一阶导很确切，也是可以理解的。
【在 l*******l 的大作中提到】
: F'(t)=int(xe^(tx)f(x)dx)怎么看出小于零？？
|
l*******l
发帖数: 248 | 15 所以总和大于零呀.不是吗？
【在 n******m 的大作中提到】
: 分正负两段积分。
: 负的那段，e^ 小于1，绝对值比 int (xfxdx) 小
: 正的那段，绝对值比 int(xfxdx) 大
: 而 int(xfxdx) 正负两段的总合是0
: 这个有点出乎意料之外，因为原不等式成立的原因是 e^x 是凸函数，但是这里我们没
: 有算二阶导。不过考虑到一阶导很确切，也是可以理解的。
|
n******m
发帖数: 169 | 16 是的， E(eX)>=eE(X)
【在 l*******l 的大作中提到】
: 所以总和大于零呀.不是吗？
|
l*******l
发帖数: 248 | 17 gotcha，搞了半天原题错了
【在 n******m 的大作中提到】
: 是的， E(eX)>=eE(X)
|
z****g
发帖数: 1978 | 18 我很愚昧的问一下第一题...大家没觉得题目有问题么....maximizes....
显然E(X)和E(Y)都大于零的话，a就是inf啊.... cov(x,y)/var(y)这类类似Beta的答案
都不对吧....是不是少个constraint.... |
z****g
发帖数: 1978 | 19 看起来是太简单了，不符合大牛的口味
【在 l*******l 的大作中提到】
: 1.why?
: 3.how did you estimate? the diff seems v big lol
: 4. what is that u don't understand, i think it is readable.
:
: t)
: To
|
z****g
发帖数: 1978 | 20 limit ( 1-1/x)^(x) = e, when x -> inf
【在 l*******l 的大作中提到】
: 1.why?
: 3.how did you estimate? the diff seems v big lol
: 4. what is that u don't understand, i think it is readable.
:
: t)
: To
|
|
|
m*******t
发帖数: 6 | 21 For #5, dont you guys know a thing called Jason's Inequality? |
z****g
发帖数: 1978 | 22 do you expect the interviewer to accept this answer?
【在 m*******t 的大作中提到】
: For #5, dont you guys know a thing called Jason's Inequality?
|
l*******l
发帖数: 248 | 23 of course, show me how to prove it using jensen, not jason...
【在 m*******t 的大作中提到】
: For #5, dont you guys know a thing called Jason's Inequality?
|
m*********g
发帖数: 646 | 24 you mean you know Jensen's inequality, you know exp() is a convex function,
you still don't get how to prove exp[E(x)]=< E[exp(x)] ?
Just take a little time, I am pretty sure you will get it.
【在 l*******l 的大作中提到】
: of course, show me how to prove it using jensen, not jason...
|
l*******l
发帖数: 248 | 25 怎么用这个估算( 1-1/x)^(x)呢？现在x=100，不是inf
【在 z****g 的大作中提到】
: limit ( 1-1/x)^(x) = e, when x -> inf
|
l*******l
发帖数: 248 | 26 这个公式哪里来的？
(1-1%)^100 binomial formula, first few terms are approximately 1-1+1/2-1/
【在 n******m 的大作中提到】
: 是的， E(eX)>=eE(X)
|
l*******l
发帖数: 248 | 27 我也觉得有问题，感觉应该是minimize var更有道理一点。
【在 z****g 的大作中提到】
: 我很愚昧的问一下第一题...大家没觉得题目有问题么....maximizes....
: 显然E(X)和E(Y)都大于零的话，a就是inf啊.... cov(x,y)/var(y)这类类似Beta的答案
: 都不对吧....是不是少个constraint....
|
l*******l
发帖数: 248 | 28 I know..that is exactly what jensen tells you. I guess the interviewer wants
you to prove jensen, not just say"Oh, it is obvious using Jensen, why do
you ask such easy question in the first place??"
,
【在 m*********g 的大作中提到】
: you mean you know Jensen's inequality, you know exp() is a convex function,
: you still don't get how to prove exp[E(x)]=< E[exp(x)] ?
: Just take a little time, I am pretty sure you will get it.
|
m*********g
发帖数: 646 | 29 google binomial coefficient formula.
I think it should be included in the beginning of your probability class.
【在 l*******l 的大作中提到】
: 这个公式哪里来的？
:
: (1-1%)^100 binomial formula, first few terms are approximately 1-1+1/2-1/
|
m*********g
发帖数: 646 | 30 Frankly, I don't think you have a very clear idea about what you are
implying here. You can proof Jensen's by convexity, it is straightforward in
this case. Just a suggestion to you (not mean but try to help), you need to
read some fundamental books. From your questions and discussions, you may
want to go over some of the basic staffs.
wants
【在 l*******l 的大作中提到】
: I know..that is exactly what jensen tells you. I guess the interviewer wants
: you to prove jensen, not just say"Oh, it is obvious using Jensen, why do
: you ask such easy question in the first place??"
:
: ,
|
|
|
l*********s
发帖数: 5409 | 31 For the purpose of job hunting,it is much more efficient to focus on the
review questions than basic theory; and, most these stuffs are irrelevant in
daily jobs.
in
to
【在 m*********g 的大作中提到】
: Frankly, I don't think you have a very clear idea about what you are
: implying here. You can proof Jensen's by convexity, it is straightforward in
: this case. Just a suggestion to you (not mean but try to help), you need to
: read some fundamental books. From your questions and discussions, you may
: want to go over some of the basic staffs.
:
:
: wants
|
m*********g
发帖数: 646 | 32 But if one missed too many basic concepts, it would be a waste of time to
focus on the "questions".
And actually not every firm like to ask stupid BT questions.
in
【在 l*********s 的大作中提到】
: For the purpose of job hunting,it is much more efficient to focus on the
: review questions than basic theory; and, most these stuffs are irrelevant in
: daily jobs.
:
:
: in
: to
|
z****g
发帖数: 1978 | 33 哦哦，公式有点错，是
lim(1+1/x)^x = e when x->inf. 数学分析里的内容，比较可以反应数轴性质的极限之
一。做过
吉米的人肯定都知道....
lim ( 1 + 1/x)^x = e when x -> inf =>
lim ( 1 - x)^(1/x) = 1/e when x->0
你再比较一下 (1-0.01)^100， 0.01和100不是随便给的，是给你凑好的
x = 0.01很接近0, 可以用这个极限近似。1/e = 0.3679
【在 l*******l 的大作中提到】
: 怎么用这个估算( 1-1/x)^(x)呢？现在x=100，不是inf
|
w******g
发帖数: 271 | 34 are you sure what you mean by 1st question? What do you mean by probably by
answering #3 ? you don't know what you're asking? |
w****i
发帖数: 143 | 35 4. The 2 year expected return is 1.05^2, standard deviation is sqrt(2)*20%.
Thus, the 1 standard deviation downward is 1.05^2-sqrt(2)*20%*norminv
again,
the
default
【在 l*******l 的大作中提到】
: 1.Find the number a which maximizes E(X+aY), where X and Y are random
: variables.
: 2. You have two fair dice, A and B. You first throw A, then B, then A again,
: etc. You stop the first time you see six. What is the probability that the
: die A will give you the first six?
: 3. Suppose you have a portfolio of 100 bonds, each with a 1% yearly default
: rate. What are the odds that no bonds in the portfolio default in a given
: year? Estimate this to the first digit. What happens if there is
: correlation between the bond defaults?
: 4.Suppose you have returns distributed with a 5% expected return and 20%
|
j*p
发帖数: 115 | 36 你知道金融课的老师怎么给学生解释的吗？
算术平均值大于等于几何平均值。
john hull那本书里好像也提到了
【在 m*******t 的大作中提到】
: For #5, dont you guys know a thing called Jason's Inequality?
|
