STUPIDSID
1 (a)
Let X be a continuous random variable with probability distribution:
Find K and P(1 ≤ X ≤ 3)
Find K and P(1 ≤ X ≤ 3)
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1 (b)
A relation R is the set of integers is defined by xRy if and only if x<y+1. Examine whether R is:
i) Reflective
ii) Symmetric
iii) Transitive
ii) Symmetric
iii) Transitive
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1 (c)
Find the eigen values and eigen vectors corresponding to following matrix:
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1 (d)
Find Laurent's series for -
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2 (a)
Seven dice are thrown 729 times. How many times do you expect at least four dice to show three or five?
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2 (b)
Evaluate the following:
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2 (c)
Show that the set of matrices
a, b ∈ z form an integral domain. Is it a field?
a, b ∈ z form an integral domain. Is it a field?
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3 (a)
Evaluate ∮c tan z dz where C is:
(i) is the circle |z|=2
(ii) is the circle |z|=1
(i) is the circle |z|=2
(ii) is the circle |z|=1
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3 (b)
"Is the following function injective, surjective?
f : R → R, f( x ) = 2x2 + 5x - 3"
f : R → R, f( x ) = 2x2 + 5x - 3"
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3 (c)
Fit a binomial distribution to the following data:
| X | 0 | 1 | 2 | 3 | 4 |
| Frequency | 12 | 66 | 109 | 59 | 10 |
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4 (a)
"If X is a normal variate with mean 10 and standard deviation 4, find:
i) P( | X - 14 | < 1)
ii) P( 5 ≤ X ≤ 18 )
iii) P( X ≤ 12)"
i) P( | X - 14 | < 1)
ii) P( 5 ≤ X ≤ 18 )
iii) P( X ≤ 12)"
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4 (b)
Let (G,*) be a group. Prove that G is an Abelian group if and only if (a * b)2 = a2 * b2.Where a2 stands for a * a.
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4 (c)
Using Poisson distribution find the approximate value of: 300C2(0.02)2(0.98)298 + 300C3(0.02)3(0.98)297.
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5 (a)
Show that the matrix
\(A=\begin{bmatrix} 1 &-6 &-4 \\0 &4 &2 \\0 &-6 &-3 \end{bmatrix}\)
is similar to a diagonal matrix. Also find the transforming matrix and the diagonal matrix
\(A=\begin{bmatrix} 1 &-6 &-4 \\0 &4 &2 \\0 &-6 &-3 \end{bmatrix}\)
is similar to a diagonal matrix. Also find the transforming matrix and the diagonal matrix
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5 (b)
A die was thrown 132 times and the following frequencies were observed:
| No. obtained : | 1 | 2 | 3 | 4 | 5 | 6 | Total |
| Frequency : | 15 | 20 | 25 | 15 | 29 | 28 | 132 |
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5 (c)
If C is a circle |z| = 1, using the integral
where K is real, show that
where K is real, show that
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6 (a)
Let A = {1, 2, 3, 5, 6, 10, 15, 30} and R be the relation 'is divisible by'. Obtain the relation matrix and draw the Hasse diagram.
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6 (b)
A certain injection administered to 12 patients resulted in the following changes of blood pressure:
5, 2, 8, -1, 3, 0, 6, -2, 1, 5, 0, 4.
Can it be concluded that the injection will be in general accompanied by an increase in blood pressure?
5, 2, 8, -1, 3, 0, 6, -2, 1, 5, 0, 4.
Can it be concluded that the injection will be in general accompanied by an increase in blood pressure?
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6 (c)
If X1 has mean 5 and variance 5, X2 has mean -2 and variance 3. If X1 and X2 are independant random variables, find -
i) E(X1 + X2), V(X1 + X2)
ii) E(2X1 + 3X2 - 5), V(2X1 + 3X2 - 5).
i) E(X1 + X2), V(X1 + X2)
ii) E(2X1 + 3X2 - 5), V(2X1 + 3X2 - 5).
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7 (a)
A random variable X has the following probability distribution:
Find:
(i) Moment Generating function
(ii) First two raw moments
(iii) First two central moments
| X: | -2 | 3 | 1 |
| P (X = x): | 1/3 | 1/2 | 1/6 |
(i) Moment Generating function
(ii) First two raw moments
(iii) First two central moments
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7 (b)
Verify Cayley Hamilton Theorem for matrix A and hence find A-1 where
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7 (c)
A random sample of 50 items gives the mean 6.2 and standard deviation 10.24. Can it be regarded as drawn from a normal population with mean 5.4 at 5% level of significance?
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