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CS/B.T
~ TECH(N)/EVEN/SEM-2/BgM-201|N}/2018-19
Matis Abd nates Ana Lines ty
of Technorsy Wes) Berg

es
Utech
a”
ee)
MAULANA ABU] KALAM AZAD UNIVERSITY OF
TECHNOLOGY, WEST BENGAL
Paper Code : BSM-201{N)
MATHEMATICS-UA
Time Allotted - 3 Hour
s Fult Marks : 70
ks.
The figures in the margin indicate full mar
dy

Candidates are required to give their answers in their own

BU MAM: dyy
Wid yNeyVur MAM

MfOoyNe
words as far as practicable.

GROUP-A
( Multiple Choice Type Questions )
correct alternatives for any ten of the
A Choose the
10x 1=10
following:
P( B) = 1/4, P( AUB) = 1/2 then
i) If P(A) = 1/3,
is —
P(B/A) 4
a) § b) 5
c} ; d} :

variable X has distribution function F.

li) The random
1/6,
tr (0)+ F(1)-+5,F(3)=§ =
3
then

P(0<X<3)}is ¢
aj 3 b) 3 Sy
¢) & gj none
of these.

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Ai) If the random variable X has Binomial distribution

with parameters n & p then the mean and variance

are respectively

<a) np & np (1-p) b) n/p&2n{1-p)

c) np(l-p)&np d) none of these.

iv) The distribution function F ( x ) of a random

eudy

ewdyy
variable X is given by
ama

MMM
wos jney

wos jney
a) P{-<KX<o) b) P(-2<Xsx)

-c) P(-xsX<wo) d) noof

ne these.

va For two random variable X and Yare independent,

then

a) E(XY}*E(XJE(Y)

b} E(XY)=B(X)+E(Y)

c) E(XY)=£(X)/
EY)

d) none of these.

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CS/B.TECH(N)/EVEN/SEM-2/BSM-201(N)/2018-19
The maximum likelihood estimate is a soluuon of
vi}
the equation

* a) MO. 9 b) GELS) — const.

c) ie =6@ d) none of these.

vii) A statistic tis said to be an unbiased estimator ofa

population parameter 8 when

b) E(t?)=8

dyY
a) E(t)=60
MM M//dyY
Woo NeyeUT

woo jneyeu
c) E(t2)=[E£(6)]? “a £(t2)=[E(t)]*.

MMA
Vai The probability of sample space is
| a) l b) 5

c) O d) none of these.

A For two events, P(A) = 04, P( B) = x and

P( At B) = 0.7. The value of x for which 4 and

B
_wiil be mutually exclusive is

—b) 3/10
a) 2/10

d) 1/5.
c) 2/5

3
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SEM-2/BSM-201 {N)/2018-19

x) The number of ways 10 persons can occupy 10

chairs arranged at a round table is

a) 101-1 b) 9-1
eh} 108 d oh
xi) The probability P( as x < b} is defined by F(x)
( where F { x) is the distribution function of the
random variable X} hutp www makaut com
ayy

a) F(b)-F(a} b) F(b})+F{a@) 2
MAM

c) F(a)-F(b) d) F(a)
F( ob). 2
NBCU

xii} A random variable X has the following p.m.f: :

x: 1 2 | 3 ‘ 5

froa:] Ff 3 fs
WOS

:
Then the value of Kis

xiii} A random variable has a Poisson distribution such

that P( 1) = P( 2). Then the s. @. of Xis

ga) 0 b} 2
c) V2 d) ~2

IT /320 1(2)(N)-2005 4.
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m
 http:/Avww.mak
aut.com
CS/B Tr CHIN}
20on
1(t,N) {2018-29
/EVEN/SEM-2/BSM-
GROUP
( sneort Answer Type_ g
Questions }
er any three of the folowi
ng. 3x5=15
2. In a bolt factory, machines
A, B and C manufacture
respectively 25%, 35% and 40% of the total out
put. 5%,
4% and 2% are defective bolts. A bolt is drawn at
random from the product and is found to be defective.

What are the probabilities that it was manufactured by

dy Yj

machines A, B and CP

AL //: dy
NEYBWU AAA

3. Find mean and variance of binomial! distribution.

Prove that ~ 1 < p xy 1, where p yy denotes correlation

MOO NEAEUT AMAL

‘coefficient between x and y.
Wo}

5. ) The distribution function F (x) of a variable Xis defined

~ as foilows :

Pl x)= A,-o< xsd!

=B-lsx<@'
awe

=C,O0sx<2

a D25xXx<@

where A, B, C, D are constant. Determine the values of

A, B, Cand D given that P(X =0)= and P(X >a)= 2,

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6. A box contains five defective and 10 non-defective

%
lamps. Eight lamps are drawn at random in succession
without replacement. What is the probability
that eighth
lamp is the fifth defective ?

GROUP -C
( Long Answer Type Questions )
Answer any three of the following. 3 x 15 = 45

What is the axiomatic definition of probability ?

dyy

MMM//:dyy
b) For two vents A and B which are nol necessarily
eur

exclusive, prove that

MMM
woo 'jney

P( AUB)=P(A)+P(B)-~P(ANB).

woo jneyew
Two bags contains respectively 3 white and 2 red

balls, 2 white and 4 red balls. One ball is drawn at

random from the first bag and is put into the

second; then a ball is drawn from the second bag.

What is the probability that the ball drawn from the

second bag is white ?

q) If A and Bare two events, prove that

P(A/B)= 1-P( A°/B), P( B)> 0.

2+5+5+3

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1{N}/2018-19

8. a) Two discrete random variable X and Y are.

connected by the relation 2X + 3¥ + 4 = 0. Prove

that the correlation coefficient between X and Y

is ~ 1,

b) Suppose that during rainy season, on a tropical

island, the length of shower has an exponential

dy Y

distribution with average length of shower 5 mins.

MALUL

day
What is the probability that a shower will last more
MOO MBYC

MMM
than three times ? [f a shower has already lasted

for 2 minutes, what is the probability that it will

wins $eyPU
last for at least one more minute ?

The joint probability density function ( pdf) of a

bivariate (X, Y} is

xs, yr O, xt y<2
f(y y)=C(xty),

=O , elsewhere

Find Cand P(X< 1, ¥> 1/2). 5+5+5

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Q a) The median and mode of the following frequency

distribution are known to be 27 and 26 respectvely

Find the values of a and b.

Class O- 10] 10-20 | 20-30 | 30-40 40-50 |

interval:

| Frequency : 3 a 20 12 b
Z
The scores of two batsman A and B in 10 innings

are:

dy
dy
ama r
woo jneyeu

A:} 19 | 31 | 48 | 53 | 67 | 90 | 10 | 62 | 40 | 80

aam//
B: | 32 | 28 | 47 | 63 | 71 | 39 | 10 | 60 | 96 | 14

wos jneyeu
Find which batsman is more consistent in scoring.

c) The radius of a circle has distribution given by the

f
probability density function :
*

f(x)=1,1l<x<2

= 0, elsewhere

/,Find the mean and variance of the area of the

circle. 5+5+5

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10. a) Fit a second degree parabola, y = a+ bx + cx?, to

the following data :

x: / O]} 1 2 3 4

y: | 1 re) 10 22 38

Predict y when x = 4.8.

A normal population has a mean 0.1 and standard

Ineyeur MMM / dy

dyy
om

AMM w
deviation 2.1. Find the probability that the mean of

woo"jARye
a sample of size 900 will be negative. Given that

P( |x|<1.43
) = 0.847.
Woo

) In order to test whether a coin is perfect the coin

tossed 5 times. The nul] hypothesis of perfectness

is rejected if more than 4 heads are obtained. What

is the probability of Type I Error ? Find the

—=, ~ re

. probability of Type Il Error when the corresponding

probability of head is 0.2. o9+9+5

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a) A machine part was designed to withstand an

11.

average pressure of 120 units. A random sample of

size 100 from a large batch was tested and it was

found that the average pressure which these parts

can withstand is 105 units with a s.d of 20 units.

Test at 5% level wheter the batch meet the

specification. Suppose the population has normal

MMAM//.dyY
amma / dy
wos jneyeur

distribution.

b) Survey of 320 families with 5 children each

wos jnNeyeu
revealed the following distribution :

No. of boys : 5 | 413 4 2 i 0

No. of girls: 8) 1 2 3 4 re)

No. of family: | 14 | 56 1110] 88 | 40 | 12

Is the result consistent with the hypothesis that

male and female births are equally probable ? The

5% value of 72 with 5 d.o.f is 11.07.

11/3201(2)(N)-2005 10
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CS/B.TECHIN) /EVEN /SEM-2/BSM-201

t of the
c} Two scanners are needed for an experimen
five available, two have electronic defects, one has

defect in memory and two are in good working

condition, Two units are selected at random. Let

X,= Number of units with electronic defects,

X= Number of units with defect in memory.

eu//-rCY
i) Find the joint distribution of X 1; and X5
too neyeur MMA: dy

no } OF Find the probabilil;

ity of no or one defects among

oo yneyMM
the selected two items.

m) ) Fi Find the marginal

distribution of x l 7

S+ 545
SS

m
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CS/B.TECH(N)/EVEN/SEM-2/2005/2023-2024/1002
“9 MAULANA ABUL KALAM AZAD UNIVERSITY OF TECHNOLOGY, WEST BENGAL
| Paper Code: BSM201 Mathematics - IIA
ae. : UPID : 002005

Time Allotted : 3 Hours Full Marks :70

, The Figures in the morgin indicate full marks. t
Candidate are required to give their answers in their own words as far as practicable

Group-A (Very Short Answer Type Question)

1. Answer any ten of the following : {1x10 10]
+f if kurtosis has a value less than 3 the distribution is called
wil if a null hypothesis accepted at 0.5 level of significance then this decision is
Wf The standard deviation of a sample mean for SRSWR |s__
4) A random variable X has the density function
f(x)= k -2<x<2
=0 otherwise
Then value of the constant kis___
lof xX has exponential distribution with parameter 1/5 then its standard deviation is
wt) if X and Y are uncorrelated then X+Y and X-Y are uncorrelated. True/False
(Vil The skew ness is positive then’meanis __—s mode
Wl tna test of hypothesis probability of Type | error is same as the level of significance of the test. True/Palse
YT in a test of hypothesis the critical region corresponding to a particular level of significance is unique. True/False
OF If Xis normally distributed with zero mean and unit variance,then the expectation of X2°
wh Normal curve of a variate represents function
~ (I) Fer X and ¥ random variables E[kf(XV}J=_

Group-B (Short Answer Type Question)

wie ~ Answer any three of the following : : + - 5x3215)
2 Marks obtained by 10 students in physics and mathematics are given in the following table. Find rank [5]
correlation between X and Y.
Marks in 48 83 40 9 16 (16 6S P24 |16 iS?
physics(X)
Marks in 13 13 24 6 15 4# 2O P 6 9
mathe matics(Y) pe “
3. Fit a straight line to the following data by the method of least squares : (5]
7 1S o bs Bo Bs ho
iy 12 4 hg 25 Bi a
<A The daily yield at a chemical plant,recorded for 50 days possesses a sample mean of 884 tons .The [5]
population standard deviation is known to be 42 tons .Do the data provide enough evidence to indicate
that the average daily yieldof the chemical plant is 900 tons per day?Test at 1% level of significance.Carry
out the test using critical value approach
In a normal distribution,31% of the items are under 45 and 8% are above 64.Find the mean and standard [5]
deviation.
[ given P{O< Z< 1.405) = 0.42 P(-0.496< Z< 0) = 0.19 J]
6. [Prove that if X and Y are independent then jolnt pmf |" [5]
s the product of corresponding marginal pmf

Group-C (Long Answer Type Question)

i Answer any three of the following : [15x3=.45 ]
i vf A bank receives on an average 2.5 customers per hour.Find the probability that in a certain hour the . [s]°
bank receivesiy'no customer = ,
ii) exactly 4 customers.Assume ‘that the number of customers received In an hour is polssonly
distributed.{ e°2> = 0.0821]?

https://www.makaut.com
 The ideal size of a class of a college is 150 students.The college,experienced from past, knows that [5].
only 30% of the admitted students will actually attend.The college uses a policy of approving the
applications of 450 students.Find the probability that more than150 students attend the class.Area
under the standard normal curve enclosed between the ordinates z=0 and z=1.59 is 0.441?
(A) A fair coin is tossed 400times .Using normal approximation ,find the probability of obtaining [5]
(i)exactly 200 heads (iijless than 210 heads (iii)between 190 and 210 heads,both inclusive .Given
that the area under standard normal curve
between z<0 and 2= 0.05 is 0.0199; between 2=0 and z= 0.95 is 0.3289
between z=0 and z= 1.05 is 0.3531
8. (a) Calculate mean, median and hence find the approximate value of the mode from the following {5]
frequency distribution:
Height(inches) 60-63 64-67 68-71 72-75 76-79 80-83
No. of Students 8 3 18 6 ‘16 8

{b) Calculate the mean deviation from the mode of the following table: [5]
Height(Inch) 60-62 63-65 66-68 69-71 72-74
Frequency 5 18 42 27 8
{c) Measurements of the lengths in feet of 50 iron rods are distributed as follows: [5]
Class boundary 2.35-2.45 2.45-2.55 2.55-2.65 2.65-2.75 2.75-2.85 2.85-2.95 2.95-3.05
frequency 1 4 7 15 11 10 2
Find the value of mean deviation upto two decimal places.

9. dat a. in order to test whether a coin is perfect the coin is tossed S times.The null hypothesis of [5]
perfectness is rejected if more than 4 heads are obtained. What is the probability of Type-! error?
Find the probability of Type-li error when the corresponding probability of head is 0.2.

sample of 28 workers is drawn from a factory to test if their average monthly wage [5]
uh b. A random
'rs.1400 or not .The population wage distribution is assumed to be normal with a standard deviation
of rs.200.The average monthly wagebased on the 25 observations comes out to be rs.1500.Perform
a test at S%level to see if the data support the claim ,using p-value approach
Gy c. To examine the claim of an expert publisher that m ,the average number of misprints per page of [5]
then the print
“a book.is 1,:if a particular page. of that book contains more’ than-2-mis null s
hypothesis that’ m=1 is rejected.Determine power of test against’ alternative hypothesis m-= 2[e
10,368] .
is rejected if 9 or more [5]
10. (a} A coin is to be tested for unbiasedness. The hypothesis that it is unbiased
e? Can you
tosses of the coin out of 10 tosses result in heads. Can we take 1% as level of significanc
suggest an appropriate level of significance for this test rule?

n respectively. The number of [5]

(b) Two samples of size 900 and 1600 are drawn from two populatio
at 1% level of significance
defective items in the two samples are 20% and 15% respectively. Test
more than that of in second
’ whether the proportion of defective items in the first population is
population, https://www.makaut.com
[given o(z>=2.33)=0.01and $(z>=1.64)=0.05
The [5]
(c) The daily yield at a chemical plant, recorded for 50 days possesses a sample mean of 884 tons
data provide enough evidence to
population standard deviation is known to be 42 tons .Do the
tons per day?Test at 1% level of
indicate that the average daily yield of the chemical plant is 900
significance.Carry out the test using p-value approach.
X denote the number of sixes and Y [5]
1. (a Consider the random experiment of throwing a pair of dice. Let
¥ and marginal pmf of X and ¥.
denote the number of fives that turn up. Find the joint pmf of X and
Also find P(X+¥2 2).
distribution of X+¥. [5]
1) if f(xty) = x+y ,0<x<1 ;O<y<1 is the joint pdf of X and Y. Find the
be the mean and standard deviation of [5]
(gf If X and Y are independent normal variate then what will
X+¥ a ?

*** END OF PAPER ***

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 CS/8 TECH(N)/EVEN/SEM 2/2005/2022 2023/1124
“eas MAULANA ABUL KALAM AZAD UNIVERSITY OF TECHNOLOGY, WEST BENGAL
fis Paper Code: BSM201 Mathematics - IIA
UPID . 002005
(Ese elt
Time Allotted . 3 Hours Full Marks 70
The Figures in the margin indicate full marks
Candidate are required to give their answers in their own words as far as practicable

Group-A (Very Short Answer Type Question)

1 Answer any ten of the following [1x 10= 10]
() jn a scatter diagram, if all the points le on a rising straight line, then {x,y} 1s said to have a
correlation
(The reyecton probability of Null Hypothesis when tt is true ts called
i) Two dice are rolled If X and ¥ denote the number appeared on first and second die, then P(X+¥=6) =

(Vl Let X be a random variable with probability distribution function f (x)=0 2 for |x|<1
=Olfor1< |x| <4
= 0 otherwise
The probabilityP(O5<x<5)is
(Vl if X and ¥ are two normal variates with standard deviations s, and sy respectively, then the standard deviation of
X+¥ is
vl) Runs scored by batsman in 5 one day matches are 50, 70, 82, 93, and 20 The standard deviation 1s
(VI The first moment about zeroisthe _ of the random variable
Vi) Mean of an exponential distribution with parameter A Is
(X) if X and ¥ are independent, then their covariance is
{%] Suppose for 40 observations, the variance is 50 If all the observations are increased by 20, the variance of these
increased observations will be
(* For a continuous random variable X, P(X=a}, where a 1s a finite number, 1s
(XI) 1f Cov(X,¥} = 0, the X and ¥ independent

Group-B (Short Answer Type Questian)

Answer any three of the following [5x3=15]
2 Fit a straight line y = a + bx to the following data [5]
xX 2 5 6 8 9
¥ 8 14 19 20 31
Predict ¥Y when X=96
3. Arandom variable X has the following pdf f(x} = k(x-1)(2-x}, for l<x<2 Find [5]
(ik (u) The distribution function F{x}
4 Iifx=4y+5 and y=kx+ 4 be two regression equatons of ‘x on y’ and ‘yon x’ respectively, then find the [5]
interval in which k les

5 From a large population, a sample of size 400 [s drawn with mean 1/71 38 Can it be reasonably regarded [5]
that the mean of the population is 17117? The sd of the population is 33 Test 5% level of significance
6 The coefficient of rank correlation coefficient of marks obtained by 10 students in English and Economics [5]
was found tobe 05 It was jater discovered that the difference in ranks in the two subjects obtained by
one of the students was wrongly taken as 3 instead of 7 Find the correct rank correlation coefficient

Group-C (Long Answer Type Question)

Answer any three of the following [i5*x3=45]

7 {a} Show that a function which 1s |x| in {-1.1) and zero elsewhere 1s a possible p d f and find the corresponding [10]
distribution function
{b} Find the mean and vanance of a continuous random variable having pd f [5]
f(x} = 1—-— |1—x|, O<x<2
8 {a} Prove that Var(X + Y} = Var(X) + Var(¥) + 2Cov{X,Y} [5]
Ve

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 (b) Ina joint distribution of XV, the marginal distributions of X and Y¥ are as follows [ 10]
X : $3 7
] l
2 2
and ¥Y : 3 6
2
My 3 3
|
If Cov(¥,Y)= >
(i) Construct the joint distribution
(if) Are X and Y independent ?
(ii) Find P(X =5,¥ =6) and P(X > Y)
3 Fit an exponential curve of the form y = ab* to the following data [15]
x 2 3 4 5 6 7
y 640 512 410 328 262 210
10 {a} From two cities, two random samples of 600 and 1000 men are drawn respectively It 1s found that [5]
400 and 600 men are illiterate among the men in the two samples respectively Test at 5% level
whether the population of the two cites have same percentage of literacy Given zy 995 = 196
(b) A company claims that the consistency regarding life time of electric bulbs produced by them is [5]
superior to those of a competitor on the basis of a study which showed that a sample of 30 bulbs
made by them hassd 25 hours while a sample of 40 bulbs of the competitor hassd 27 hours Test
at 5% level whether the claim of the company is justified Given zg 95 = 1 645
There are two normal populations, | and Ul, having sd 8 and & respechvely Two independent [5]
samples of size 10 and 12 are drawn from the two populations respectively with sample mean 20
and 27 Test at 1% level whether the two population means are equal Given Zgqgs = 2 58
The length of bolts produced by a machine is nonnally distnbuted with mean 4 ands d 04 A bolt is defecnveif its [6]
length does not he mi the mterval(} § 43) Findthe percentage of defect c bolts produced by the machine Gren
ce f
1 - 77 ace 1 frst
Trl dt =0 7257 Tre? dr =0 6554

(b) lf the weekly wages of 10,000 workers in a factory follows normal distribution with mean and [9]
standard deviation Rs 70 and Rs 5 respectively then find the expected number of workers whose
weekly wages are https (www makaut com
(ij between Rs 66 and Rs 72 (ih lessthanRs 66 (ii) more than Rs 72
Given that the area under the standard normal curve between z= 0 and z= 041501554 andz=0
andz=081s
0 2881

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