Probability and Statistics (March-2023)

Maulana Azad National Urdu University M.Sc. (Maths) I Semester Examination, March 2023

MSMM104CCT : Probability and Statistics

È Z w Zz Ñ g c g

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Time : 3 hrs Marks : 70

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” x Ð V Z Î V ß

» Z Ž

[ X ì è S g  á Z Š Z ® Å V / n Æ [ Z Ž C Ù X x Î z Ô zx Š z Ô w z Zz : ì Œ 6 , V ” & ] Ñ Z Î p ! 6 , t

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~ w Zz z

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(10 x 1 = 10 Marks)

X ì – 

¹

a

/ 6 V ,

Œ

⁽²⁰⁰⁾

Î z Š ã ½ [ Z Ž » w Z Î C Ù X  ¶ Š [ Z Ž Æ V ß Z Î õ 0 * ð à à D ¨ ( ¤ Ð ~ k XZ  ] Ñ Z Î J W ~ zx Š z

^.2

(5 x 6 = 30 Marks)

X  – ] Z 

⁶

a Æ w Z Î C Ù X ì

V / 6 ,

Œ

⁽⁵⁰⁰⁾

Î õ 0 * ã ½ [ Z Ž » w Z Î C Ù X  ¶ Š [ Z Ž Æ V ß Z Î & ð à à D ¨ ( ¤ Ð ~ k XZ  ] Ñ Z Î õ 0 * ~ x Î z

^.3

(3 x 10 = 30 Marks)

X  – ] Z 

¹⁰

a Æ w Z Î C Ù X ì

Zz z w

1 :

 w Z Î

ƒ A

$ ¤ / Z

⁽ⁱ⁾

Z

~ y Ð ð Ã

7

^{(d) (c) (b) (a)}

Randomly

à V Y ±

⁽²⁾

z Š ¤ / XZ 

^(Blue)

š @ M Å

⁽³⁾

& Ð ~ X  V H ±

⁽¹⁰⁾

k Š ~

^(Class)

® ) ) q - Z

⁽ⁱⁱ⁾

ƒ

ž Ï

Š â z Å V M

š @

ƒ ' ' '

'

Probability

H 6 , ä ™

^Select

Z

~ y Ð ð Ã

7

^{(d) (c) (b) (a)}

7

(

^/

V ; ) ? 7 t  @ * ƒ ' , Z ' ,

^variance

g Zz

^mean

~

Binomial distribution

H

⁽ⁱⁱⁱ⁾

ƒ

X Ç

^mean

A $

P(x=1)=P(x=2)

ž b § k Z ì

Poission Variate

q - Z

^X

¤ / Z

^(iv)

Z

~ y Ð ð Ã

7

^{(d) 3 (c) 2 (b) 1 (a)}

ƒ

X Ç

^{.... = n}

A $ ƒ ~

Binomial Distribution

¤ / Z

^(v)

Z

~ y Ð ð Ã

7

^{(d) 15 (c) 12 (b) 10 (a)}

1/4

E Z

™ w D



X

^test

X X X X X X X ë A $ ƒ N * g

sample size

¤ / Z

^(vi)

-test- (d) F-test (c) t-test (b) Z-test (a)

ƒ Ð

X X X X X X X

sample space

n Z A $ ƒ Š H Ñ Y Z û %

¹⁰

Ã

^(Coin)

ö q - Z

^(vii)

Z

~ y Ð ð Ã

7

^{(d) (c) (b) (a)}

Š c

*

Š ì H A ë $ E Z

™ w D

 ' ' ' '

'

^mean

z Š ~

Normaldistribution

¤ / Z

^(viii)

Z

~ y Ð ð Ã

7

^{(d) (c) (b) (a)}

' ~

' ' ' '

'

Normal Distribution (ix)

mean < median < mode (b) mean = median = mode (a)

Z

~ y Ð ð Ã

7

^(d) mean > median > mode(c)

(x)

Z

~ y Ð ð Ã

7

^{(d) (c) (b) (a)}

Š z

zx X z ™ " $ U * g Zz y Ò Ã Bayes Theorem (2)

2%, 3%, 6%

Ð ~ T  D ¯

bulb 50% C

g Zz

^{30% B}

ó

20% A Machines

Æ ~ 1

Electric Bulb

q - Z

⁽³⁾

ƒ

ž Ï

z{

Probability

H X 

^Defective

z{ ž Z ƒ x ¥ g Zz  ä 3

^Randomly

Ã

^Bulb

q - XZ 

Defective Bulb

¯ C



X

C,B,A Machines

ó

^Bulb

Z

§ k

Š b c

*

Š ì H

X

Probability Density Function (p.d.f)

»

Continuous Random Variable

q - Z

⁽⁴⁾

Variance (iii) Mean (ii) K (i)

z ™ x ¥ A $

¥

™ x z

X

Probability

A $ X 

^{S.D = 16}

g Zz

^{Mean = 70}

» T 

Normal Variate

q - Z

^X

¤ / Z

⁽⁵⁾

(ii) (i)

Æ

! g *

~ }

Ÿ ÉÀ /õG

X

Critical Region

g Zz

Type I, Type II Error (6)

2/4

% Æ ä

Š - z E - g Ð V a Zz ‰ gZ Æ y zi Ð y

³

¤ Zx / Zz

g

(students)

Y C

⁽⁷⁾

X

Test the significance of the difference between the means

X  Š H c * Î [ ˆ »

Standard Deviation

Mean S.D Size of Sample

University A 55 10 400

University B 57 15 100

(Table Value of Normal distribution )

4000 hrs

Å

Average life time electric bulb

¤ / XZ ì ˆ ~ Š ~

^Data

n

^{Life Time}

Å

10 Electric Bulbs (8)

™ M

 h

X

^Accept

Ã

^Hypothesis

ë H A $ ì

Items 1 2 3 4 5 6 7 8 9 10

Life of 1000hrs 1.2 4.6 3.9 4.1 5.2 3.8 3.9 4.3 4.4 5.6

(

Table value

ò

Š c

*

Š ì H

X

^Data

»

daily wages

~ 9 gz Å

skilled workers

~ V z à z Š

⁽⁹⁾

City Size Standard Deviatin of wages in the sample

City A 16 25

City B 13 12

™ ,

X

^Test

Ã

Variances of the wages distributed in the two cities (Table value F_0.05 (12,15)=2.48)

z x Î

U

*

"

™ $

ž z

:

⁽¹⁰⁾

(ii) (i)

X z ™ x ¥ g Š Å A $ ì Š H c * Š

^p.d.f

~

Normal Distribution (11)

a Æ

^success

z Š A $ ì

^Variance=2

g Zz

^Mean=6

»

Binomial Distribution(i) (12)

¥

™ x z

X

probability

ì Z

§ k

b

poission variate

q - Z

" X"

¤ / Z

⁽ⁱⁱ⁾

(b) Mean (a)

z ™ x ¥ A $

3/4

ì

^{S.D = 1.6}

g Zz

Mean = 31.45

» T 6 ,

40 Diesel engines

ì Š H H

test perform

q - Z

⁽¹³⁾

ì

X

null hypothesis against the alternate hypothesis

H ž , ™ õ Y

(Table value of normal distribution )

X ì Š H c * Š

according to sex and nature of work

»

^{agency (2)}

z Š

⁽¹⁴⁾

ì

c 7 *

?

nature of work independent of sex workers

H ž , ™ G F +B/õE Y

sex Agency 1 Agency 2 Total

Male 40 20 60

Female 10 30 40

Total 50 50 100

(Table Value )

/ / /

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