Sample Questions of ISI ME I (Mathematics) 2010 Discussion

[Benhur]

the ans to d 1st part..will it be 0??
since F(X) is integration of d values of X,the expectation of F(X) will be sum of expectation of the random variable till X..which is sum of zeros..

is it correct??

[sonal]

sir i got the answer 0 according to the following method...
var(x(N))
=var(x1/n+x2/n+......+xn/n)
=var(x1/n)+var(x2/n)+.......+var(xn/n)+2cov(x1/n,x2/n)+2cov(x2/n,x3/n)+....+2cov(xn/n,x1/n)
=n(sq)s/(sq)n+2np/(sq)n
=(sq)s/n+2p/n
lim as n tends to infinity var(x(n)) will therefore be 0.

sir please point out the mistake i'm making............
thank u

[Amit Goyal]

Hi Sonal, mistake is quoted in the following step
=var(x1/n)+var(x2/n)+.......+var(xn/n)+"2cov(x1/n,x2/n)+2cov(x2/n,x3/n)+....+2cov(xn/n,x1/n)"
according to what you wrote (as quoted) there will be n covariance type terms but thats not the case and there are (sq(n) - n)/2 such terms.
For example when n = 4 then we have (sq(n) - n)/2 = 6 terms and not 4. Here is the elaboration:
=var(x1/n)+var(x2/n)+var(x3/n)+var(x4/n)+2cov(x1/n,x2/n)+2cov(x1/n,x3/n)+2cov(x1/n,x4/n)+2cov(x2/n,x3/n)+2cov(x2/n,x4/n)+2cov(x3/n,x4/n)

[sonal]

thank u sir:)

[Amit Goyal]

Solutions are as follows:
Let X be a Normally distributed random variable with mean 0 and variance 1. Let F(.) be the cumulative distribution function of the variable X. Then the expectation of F(X) is
(c) 1/2
(Hint: F(X) has uniform distribution on (0, 1))

Consider any finite integer r ≥ 2. Then lim(x→0) f(r, x)/a(x) equals,
(where f(r, x) = log(e)(Σ(0, r) (x^k)); and a(x) = Σ(1, ∞) ((x^k)/k!))
(b) 1

Note: Σ(a, b) g(k) is summation of g(k) over values of k from a to b
x^k is x to the power k

[Amit Goyal]

Next one:

Consider 5 boxes, each containing 6 balls labelled 1, 2, 3, 4, 5, 6. Suppose one ball is drawn from each of the boxes. Denote by b(i), the label of the ball drawn from the i-th box, i = 1, 2, 3, 4, 5. Then the number of ways in which the balls can be chosen such that b(1) < b(2) < b(3) < b(4) < b(5) is
(a) 1,
(b) 2,
(c) 5,
(d) 6.

[priyanka p]

it should be option (b) 2

[Benhur]

d no of ways are..

1 2 3 4 5
1 2 3 4 6
1 2 3 5 6
1 2 4 5 6
1 3 4 5 6
2 3 4 5 6

so d ans is 6..rite??

[Amit Goyal]

That's right.

Consider 5 boxes, each containing 6 balls labelled 1, 2, 3, 4, 5, 6. Suppose one ball is drawn from each of the boxes. Denote by b(i), the label of the ball drawn from the i-th box, i = 1, 2, 3, 4, 5. Then the number of ways in which the balls can be chosen such that b(1) < b(2) < b(3) < b(4) < b(5) is
(d) 6.

[Benhur]

sir can u explain y F(X) has an uniform dist??

[Amit Goyal]

First of all, F(X) take values in the interval (0, 1) because F(.) is a cumulative distribution function. So, for 0 < a < 1 we have:
P(F(X) < a) = P(X < Finv(a)) (Since F is strictly increasing)
                = F(Finv(a)) = a
This implies that F(X) is distributed uniformly because Probability that F(X) take values less than a = a ∀ a ∈ (0, 1)

[Amit Goyal]

The sum C(n+0, 0) + C(n+1, 1) +............C(n+m, m)equals
(a) C(n+m+1, n+m),
(b) (n+m+1)C(n+m, n+1),
(c) C(n+m+1, n),
(d) C(n+m+1, n+1).

Consider the following 2-variable linear regression where the error e(i)’s are independently and identically distributed with mean 0 and variance 1; y(i) = a + b(x(i) − Mean(x)) + e(i), i = 1, 2, . . . , n. where Mean(x) = (x(1) + x(2) + ....x(n))/n
Let a^ and b^ be ordinary least squares estimates of a and b respectively.
Then the correlation coefficient between a^ and b^ is
(a) 1,
(b) 0,
(c) −1,
(d) 1/2.

[Benhur]

the ans to d 2nd questn is
-1

[priyanka p]

ans 1 is option (b)  and 2nd answer is (c) -1

[beni.bear02]

can u plz xplain d 1st ques??

[priyanka p]

actually m sorry, it shuld b option a

[priyanka p]

i try 2 give an explanation, expanding the binomial coefficients,    we get Ln/Ln + L(n+1)/Ln + L(n+2)/Ln L2 +-------------------+ L(n+m)/Ln Lm    ; whereLn means factorial n  solving we get 1+ (n+1) + {(n+1)(n+2)/L2} +--------------+[{(n+1)(n+2)(n+3)---(n+m)}/ Lm]  therefore we get option a as  solution

[Chandrani]

@ priyanka
Hi.. i got till the second last part.. where you get the simplified form of the coefficients.. but how does that imply the ans as (a).. could you xplain..

[Varsha]

The function f(x) = x(square_root(x) + square_root(x + 9)) is
(a) continuously differentiable at x = 0,
(b) continuous but not differentiable at x = 0,
(c) differentiable but the derivative is not continuous at x = 0,
(d) not differentiable at x = 0.
Note: square_root(g(x)) stands for square root of g(x)


Sir, cud u explain hw u got it as continuously differentiable at x = 0

[Amit Goyal]

Find the derivative of the function. Its easy to see that it exists. Notice that derivative is also a function, check for its continuity.