# Sample Questions of ISI ME I (Mathematics) 2010 Discussion Classic List Threaded 128 messages Open this post in threaded view
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 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??
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 In reply to this post by Amit Goyal 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
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 Administrator 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)
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 thank u sir:)
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 Administrator In reply to this post by 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
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 Administrator 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.
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 it should be option (b) 2
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 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??
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 Administrator In reply to this post by 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.
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 In reply to this post by Benhur sir can u explain y F(X) has an uniform dist??
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 Administrator 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)
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 Administrator In reply to this post by 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.
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 the ans to d 2nd questn is -1
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 In reply to this post by Amit Goyal ans 1 is option (b)  and 2nd answer is (c) -1
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 can u plz xplain d 1st ques??
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 actually m sorry, it shuld b option a
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 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
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## Re: Sample Questions of ISI ME I (Mathematics) 2010 Discussion

 @ 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..