1 (a)
Let X be a continuous random variable with probability distribution:
![](data:image/png;base64,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)
Find K and P(1 ≤ X ≤ 3)
Find K and P(1 ≤ X ≤ 3)
5 M
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
5 M
1 (c)
Find the eigen values and eigen vectors corresponding to following matrix:
5 M
1 (d)
Find Laurent's series for -
5 M
2 (a)
Seven dice are thrown 729 times. How many times do you expect at least four dice to show three or five?
7 M
2 (b)
Evaluate the following:
7 M
2 (c)
Show that the set of matrices
![](data:image/png;base64,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)
a, b ∈ z form an integral domain. Is it a field?
a, b ∈ z form an integral domain. Is it a field?
6 M
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
7 M
3 (b)
"Is the following function injective, surjective?
f : R → R, f( x ) = 2x2 + 5x - 3"
f : R → R, f( x ) = 2x2 + 5x - 3"
7 M
3 (c)
Fit a binomial distribution to the following data:
X | 0 | 1 | 2 | 3 | 4 |
Frequency | 12 | 66 | 109 | 59 | 10 |
6 M
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)"
7 M
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.
7 M
4 (c)
Using Poisson distribution find the approximate value of: 300C2(0.02)2(0.98)298 + 300C3(0.02)3(0.98)297.
6 M
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
7 M
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 |
7 M
5 (c)
If C is a circle |z| = 1, using the integral
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where K is real, show that
where K is real, show that
6 M
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.
7 M
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?
7 M
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).
6 M
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
7 M
7 (b)
Verify Cayley Hamilton Theorem for matrix A and hence find A-1 where
7 M
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?
6 M
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