ISI 2010

[Manvendra]

Hi,

Please help me in the following questions :

1.

Let X be a Normally distributed random variable with mean 0 and variance 1. Let Y be the cumulative distribution function of the variable X . Then the expectation of Y is .



2 .

Let Xi , i = 1 ,2 , 3 , 4 .. n be identically distributed with variance *sigma*^2 . .
Let cov(Xi,Xj) = m for all i is not equal to j. Define Y = 1/n Summation Xi and let an = Variance ( Y )
Then limit n tends to infinity an equals .


Thanks in advance .

[duck]

Hi Manvendra.. :)

For Q1) Refer Pg 6 of the following link>
http://economicsentrance.weebly.com/uploads/1/1/0/5/1105777/notes_1.pdf

Q2) First find V(Y)
V(Y) = V(X1+....+Xn/n) = 1/n^2 [ V(X1+....+Xn)]
                                = 1/n^2 [ V(X1) + V(X2)+....+V(Xn) + (n)(n-1) Cov(Xi,Xj) ]
Note: As there are "n" Xi , So the total number of ways in which we can select two Xi is [nC2 = (n)(n-1)/2]. Therefore, there are Cov(Xi,Xj) is added (n)(n-1)/2 times. Substitute this in formula of calculating Variance.

Now we have, V(Y) = [n*(sigma^2) + (n)(n-1)(m)]/n^2
Find the limit of V(Y) as n tends to infinity, you'll get "m"

[ritu]

Hello duck.
Pls pls tell me solution to question 8 of isi 2011 Me2....solow model one...i really want to kno how to do that question...its quite long so havent posted it...

[duck]

Hi Ritu.. :)

Q8) Part(a)
Y= sqrt(KL)
=> f(k) = sqrt(k)  where k = K/L (capital-output ratio)
At steady state:
s*f(k) = k*n
=> 0.20*sqrt(k) = k*0.05
=> k = 16
So, we have capital output ratio = 16.

For rate of growth of output>
Y= L*f(k)
take log and differentiate, you'll get:
At Steady state, rate of growth of output = rate of growth of labour = 0.05
[Note: Its coming out to be independent of savings]

For Rate of growth of savings:
S= s*Y
Again take log and differentiate, you'll get:
At steady state, Rate of growth of Savings = Rate of growth of output
Therefore, Rate of growth of Saving = Rate of growth of labour = 0.05

For wage rate>
As we assume pefectly Competitve markets, so wages = Marginal product of labour
So, you'll get w=2

Part(b)
As we found out that rate of growth of putput is independent of savings. So, an increase in savings will have no effect on the growth rate of output.

Part(c)
In solow model, savings is "exogenous" and it only leads changes in the level and "not" the growth rates.

[ritu]

Hey duck thanx a lot...but ek choti si prob h...i askd for question 8 of ""2011"""u told me questn 8 of 2010...:-P:-P
Sry for the inconvenience....:-(

[Manvendra]

Hi

thanks for the reply.

just one more doubt ..the var(x + y) = var x + var y + 2 cov ( x,y ) or is it var x + var y - 2 cov (x,y) ?


thanks in advance

[duck]

Hi.. :)

V(X+Y) = V(X)+V(Y)+2Cov(X,Y)

V(X-Y) = V(X)+V(Y)-2Cov(X,Y)

[duck]

@Ritu:
hehe..sorry!!
will post it as soon as i will do it.. :)

[ritu]

yes duck...i badly need to know it...thank:)