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 ∮

(i) is the circle |z|=2

(ii) is the circle |z|=1

_{c}tan z dz where C is:(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 ) = 2x

f : R → R, f( x ) = 2x

^{2}+ 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}= a^{2}* b^{2}.Where a^{2}stands for a * a.
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4 (c)
Using Poisson distribution find the approximate value of:

^{300}C_{2}(0.02)^{2}(0.98)^{298}+^{300}C_{3}(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 X

i) E(X

ii) E(2X

_{1}has mean 5 and variance 5, X_{2}has mean -2 and variance 3. If X_{1}and X_{2}are independant random variables, find -i) E(X

_{1}+ X_{2}), V(X_{1}+ X_{2})ii) E(2X

_{1}+ 3X_{2}- 5), V(2X_{1}+ 3X_{2}- 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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