STUPIDSID
1 (a)
A fair coin is tossed till head appears. What is the expectation of the number of tosses required?
5 M
1 (b)
Solve using Bisection method
x - cosx = 0. Find the positive root.
x - cosx = 0. Find the positive root.
5 M
1 (c)
Solve graphically the following L.P.P.
Maximize, z = x - 2y
Subject to, -x + y ≤ 1
6x + 4y ≥ 24
0 ≤ x ≤ 5, 2 ≤ y ≤ 4
Maximize, z = x - 2y
Subject to, -x + y ≤ 1
6x + 4y ≥ 24
0 ≤ x ≤ 5, 2 ≤ y ≤ 4
5 M
1 (d)
The mean value of random sample of 60 items was found to be 145 with standard deviation of 40. Find the 95% confidence limits for the population mean.
5 M
2 (a)
If p.d.f. of a random variable is given by,
f(x) = x for 0 ≤ x ≤ 1
f(x)=2 - x for 1 ≤ x ≤ 2
f(x)=0 Otherwise
Find the m.g.f. and hence find mean and variance.
f(x) = x for 0 ≤ x ≤ 1
f(x)=2 - x for 1 ≤ x ≤ 2
f(x)=0 Otherwise
Find the m.g.f. and hence find mean and variance.
6 M
2 (b)
If x1 and x2 are independent normal variates with mean 30 and 25 and variance 16 and 12 respectively and y = 3x1 -x2. Find P(60 ≤ y ≤ 80)
6 M
2 (c)
Evaluate
(i) Trapezoidal Rule.
(ii) Simpons 1/3rd rule.
(iii) Simpons 3/8th rule.
Take h = 0.25. Compare the results with exact value.
(i) Trapezoidal Rule.
(ii) Simpons 1/3rd rule.
(iii) Simpons 3/8th rule.
Take h = 0.25. Compare the results with exact value.
8 M
3 (a)
Test significance of difference between the means of samples drawn from two normal populations with the following data:
| Size | Mean | s.d. | |
| Sample I | 100 | 61 | 4 |
| Sample II | 200 | 63 | 6 |
6 M
3 (b)
If x = au + b, y = cv + d; a, b, c, d are constants then prove rxy=ruv
where rxy is coefficient of correlation between x and y.
where rxy is coefficient of correlation between x and y.
6 M
3 (c)
Fit a second degree curve for the following data:
| x | 1 | 2 | 3 | 4 | 5 |
| y | 1250 | 1400 | 1650 | 1950 | 2300 |
8 M
4 (a)
Using Gauss-Seidel method, solve the equations
10x + y + z = 12;
2x + 10y + z = 13
2x + 2y + 10z = 14
10x + y + z = 12;
2x + 10y + z = 13
2x + 2y + 10z = 14
6 M
4 (b)
According to theory of proportion of commodity in the four classes A, B, C, D should be 9 : 2 : 4 : 1. In a survey of 1600 items of this commodity the numbers in four classes were 882, 432, 168 and 118. Does this survey support the theory?
6 M
4 (c)
Find the coefficient of correlation for the following data:
| x | 2 | 4 | 5 | 6 | 8 | 11 |
| y | 18 | 12 | 10 | 8 | 7 | 5 |
8 M
5 (a)
Let X be a random variable with the following p.d.f.
Find the mean, variance and E(2X+1)2.
| x | -3 | 6 | 9 |
| P(x) | 1/6 | 1/2 | 1/3 |
Find the mean, variance and E(2X+1)2.
6 M
5 (b)
Explain:-
(i) Null Hypothesis.
(ii) Alternate Hypothesis.
(iii) Critical Region.
(iv) Level of Significance.
(v) Types of errors.
(vi) One Tailed Two Tailed Tests.
(i) Null Hypothesis.
(ii) Alternate Hypothesis.
(iii) Critical Region.
(iv) Level of Significance.
(v) Types of errors.
(vi) One Tailed Two Tailed Tests.
6 M
5 (c)
Find f(8) from the data -
| x | 5 | 7 | 11 | 13 | 17 |
| f (x) | 150 | 392 | 1452 | 2366 | 5202 |
8 M
6 (a)
Solve using Gauss-Jordan method
2x + y + 4z = 16;
3x + 2y + z = 10
1x + 3y + 3z = 16.
2x + y + 4z = 16;
3x + 2y + z = 10
1x + 3y + 3z = 16.
6 M
6 (b)
How many tosses of a coin are needed so that the probability of getting at least one head is 87.5%?
6 M
6 (c)
Solve:
Maximize: z = 4x1 +x2 + 3x3 + 5x4;
Subject to, 4x1 - 6x2 - 5x3 - x4 ≤ 2
-3x1 - 2x2 + 4x3 + x4 ≤ 10
-8x1 - 3x2 + 3x3 +2x4 ≤ 20
x1 ,x2, x3, x4 ≥ 0.
Maximize: z = 4x1 +x2 + 3x3 + 5x4;
Subject to, 4x1 - 6x2 - 5x3 - x4 ≤ 2
-3x1 - 2x2 + 4x3 + x4 ≤ 10
-8x1 - 3x2 + 3x3 +2x4 ≤ 20
x1 ,x2, x3, x4 ≥ 0.
8 M
7 (a)
Find mean and variance of Binomial Distribution.
6 M
7 (b)
Two batches of 12 animals are taken for test of inoculation. One batch was inoculated and other was not from the data can it be regarded as effective against the disease?
| Dead | Survived | Total | |
| Inoculated | 2 | 10 | 12 |
| Non-Inoculated | 8 | 4 | 12 |
| Total | 10 | 14 | 24 |
6 M
7 (c)
Show that R=r for the following data:
| x | 60 | 62 | 64 | 66 | 68 | 70 | 72 | 74 |
| y | 92 | 83 | 101 | 110 | 128 | 119 | 137 | 146 |
8 M
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