Problem: A bank issues credit cards to customers under the scheme of Master Card. Based on the past data,the bank has found out that 60% of all accounts pay on time following the bill. If a sample of 7 accounts is selected at random from the database, construct the Binomial Distribution of accounts paying on time?

Random Variable: x = number of accounts paying on time, 1-7

p = 0.6Sample Size = 7

Binomial Distribution

No.Accounts Probability Cumulative Probability0 0.0016384 0.00163841 0.0172032 0.01884162 0.0774144 0.0962563 0.193536 0.2897924 0.290304 0.5800965 0.2612736 0.84136966 0.1306368 0.97200647 0.0279936 1

Mean 4.2Standard deviation 1.3

1 2 3 4 5 6 7 8

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

P( x )=( nx ) px (1−p )(n− x )

1 2 3 4 5 6 7 8

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1

1.2

P( x )=( nx ) px (1−p )(n− x )

Proble: If on an average, 3 customers arrive every one minutes at a bank during the busy hours of working,(a) what is the probability that exatly four customers arrive in a given minute?(b) what is the probability that more than three customers arrive in a given minute?

Mean = 3

No.of Customers Probability Cumulative Probability0 0.0497871 0.04978706836786391 0.1493612 0.1991482734714562 0.2240418 0.4231900811268443 0.2240418 0.6472318887822314 0.1680314 0.8152632445237725 0.1008188 0.9160820579686976 0.0504094 0.9664914646911597 0.021604 0.9880954961436438 0.0081015 0.9961970079383249 0.0027005 0.998897511869884 Mean

10 0.0008102 0.999707663049353 Standard deviation

1 2 3 4 5 6 7 8 9 10 11

0

0.05

0.1

0.15

0.2

0.25

P( x )= e− λ λx

x !

Proble: If on an average, 3 customers arrive every one minutes at a bank during the busy hours of working,

31.73

1 2 3 4 5 6 7 8 9 10 11

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0.05

0.1

0.15

0.2

0.25

1 2 3 4 5 6 7 8 9 10 11

0

0.2

0.4

0.6

0.8

1

1.2

P( x )= e− λ λx

x !

Problem: The mean weight of a morning breakfast cereal pack is 0.295 kg with a standard deviation of 0.025 kg.The random variable weight of the pack follows a normal distribution. Answer the following questions:(1) What is the probability that the pack weighs less than 0.280 kg? 0.274253 27.43(2) What is the probability that the pack weighs more than 0.350 kg? 0.013903 1.39(3) What is the probability that the pack weighs between 0.260 to 0.340? 0.883313 88.33

Mean 0.295Standard deviation 0.025

x Z valueProbability Cumulative Probability0.16 -5.4 7.43E-06 3.3320448485429E-08

0.175 -4.8 0.000158 7.9332815197559E-070.19 -4.2 0.002358 1.3345749015906E-05

0.205 -3.6 0.024476 0.000159108590157530.22 -3 0.177274 0.0013498980316301

0.235 -2.4 0.895781 0.008197535924596130.25 -1.8 3.158006 0.0359303191129258

0.265 -1.2 7.767442 0.1150696702217080.28 -0.6 13.32898 0.274253117750074

0.295 0 15.95769 0.50.31 0.6 13.32898 0.725746882249927

0.325 1.2 7.767442 0.8849303297782920.34 1.8 3.158006 0.964069680887074

0.355 2.4 0.895781 0.9918024640754040.37 3 0.177274 0.99865010196837

0.385 3.6 0.024476 0.9998408914098420.4 4.2 0.002358 0.999986654250984

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

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4

6

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10

12

14

16

18

f ( y )= 1√2πσ

e− 1

2σ2( y−μ )2

Problem: The mean weight of a morning breakfast cereal pack is 0.295 kg with a standard deviation of 0.025 kg.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0

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4

6

8

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12

14

16

18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0

0.2

0.4

0.6

0.8

1

1.2

f ( y )= 1√2πσ

e− 1

2σ2( y−μ )2

Problem: If breakdowns in an equipment occur completely at random, with an average interval between breakdowns of 10 months,What is the probability that the next breakdown will occur in 6 months or less?

MTBF 10Lambda 0.10

No.of Breakdowns Probability Cumulative Probability0 0.1 01 0.090484 0.09516258196404052 0.081873 0.1812692469220183 0.074082 0.2591817793182824 0.067032 0.3296799539643615 0.060653 0.3934693402873676 0.054881 0.4511883639059747 0.049659 0.5034146962085918 0.044933 0.5506710358827789 0.040657 0.593430340259401

10 0.036788 0.63212055882855811 0.033287 0.66712891630192112 0.030119 0.69880578808779813 0.027253 0.72746820696598814 0.02466 0.75340303605839415 0.022313 0.7768698398515716 0.02019 0.79810348200534517 0.018268 0.81731647594726518 0.01653 0.83470111177841319 0.014957 0.85043138077736520 0.013534 0.86466471676338721 0.012246 0.87754357174701822 0.01108 0.88919684163766623 0.010026 0.899741156277196

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

0.02

0.04

0.06

0.08

0.1

0.12

Problem: If breakdowns in an equipment occur completely at random, with an average interval between breakdowns of 10 months,

Mean 10Standard deviation 10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

0.02

0.04

0.06

0.08

0.1

0.12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

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0.2

0.4

0.6

0.8

1

1.2

f ( y )=1θe− yθ

Mean0.011017 0.709098 0.620106 0.775445 0.433058 0.5097450.223609 0.658773 0.14127 0.204382 0.359416 0.317490.731773 0.990661 0.921751 0.999023 0.674612 0.8635640.868435 0.296579 0.469192 0.622517 0.521683 0.5556810.129124 0.744896 0.024262 0.983398 0.682394 0.5128150.451613 0.259621 0.845393 0.48793 0.573229 0.5235570.141209 0.877194 0.114658 0.270821 0.082797 0.2973360.646748 0.931516 0.719291 0.214423 0.299295 0.5622550.395672 0.367656 0.226569 0.150884 0.897641 0.4076850.529649 0.223518 0.252297 0.693594 0.710379 0.4818870.368175 0.580187 0.95761 0.575701 0.598926 0.616120.011292 0.483718 0.970489 0.200049 0.736198 0.4803490.069979 0.891842 0.507004 0.268563 0.115146 0.3705070.823603 0.527543 0.706717 0.459639 0.338664 0.5712330.760277 0.402997 0.001953 0.375011 0.991577 0.506363 00.697867 0.121525 0.091525 0.352489 0.87463 0.427607 0.10.439711 0.710746 0.416334 0.991089 0.949675 0.701511 0.20.253517 0.618885 0.744285 0.046846 0.573077 0.447322 0.30.781854 0.025056 0.491012 0.593829 0.021241 0.382598 0.40.317942 0.831111 0.161534 0.821741 0.603076 0.547081 0.50.324412 0.411176 0.558153 0.91644 0.880917 0.61822 0.60.058504 0.245735 0.224555 0.064425 0.521867 0.223017 0.70.799982 0.769829 0.681722 0.280129 0.587512 0.623835 0.80.542619 0.990417 0.373302 0.549272 0.037019 0.498526 0.90.057588 0.45204 0.677206 0.945677 0.300089 0.48652 10.282052 0.125675 0.950377 0.737419 0.110111 0.4411270.739097 0.264718 0.988922 0.797327 0.276345 0.6132820.084994 0.546434 0.126377 0.114322 0.605274 0.295480.353526 0.189032 0.710807 0.753533 0.222571 0.445894

0.25959 0.608509 0.304422 0.121006 0.061281 0.2709620.011017 0.709098 0.620106 0.775445 0.433058 0.5097450.223609 0.658773 0.14127 0.204382 0.359416 0.317490.731773 0.990661 0.921751 0.999023 0.674612 0.8635640.868435 0.296579 0.469192 0.622517 0.521683 0.5556810.129124 0.744896 0.024262 0.983398 0.682394 0.5128150.451613 0.259621 0.845393 0.48793 0.573229 0.5235570.141209 0.877194 0.114658 0.270821 0.082797 0.2973360.646748 0.931516 0.719291 0.214423 0.299295 0.5622550.395672 0.367656 0.226569 0.150884 0.897641 0.4076850.529649 0.223518 0.252297 0.693594 0.710379 0.4818870.368175 0.580187 0.95761 0.575701 0.598926 0.616120.011292 0.483718 0.970489 0.200049 0.736198 0.4803490.069979 0.891842 0.507004 0.268563 0.115146 0.3705070.823603 0.527543 0.706717 0.459639 0.338664 0.5712330.760277 0.402997 0.001953 0.375011 0.991577 0.5063630.697867 0.121525 0.091525 0.352489 0.87463 0.4276070.439711 0.710746 0.416334 0.991089 0.949675 0.7015110.253517 0.618885 0.744285 0.046846 0.573077 0.4473220.781854 0.025056 0.491012 0.593829 0.021241 0.3825980.317942 0.831111 0.161534 0.821741 0.603076 0.547081

Random Numbers

Random Numbers

Random Numbers

Random Numbers

Random Numbers

1 2 3 4 5 6 7 8 9 10

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10

20

30

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60

70

80

0.324412 0.411176 0.558153 0.91644 0.880917 0.618220.058504 0.245735 0.224555 0.064425 0.521867 0.2230170.799982 0.769829 0.681722 0.280129 0.587512 0.6238350.542619 0.990417 0.373302 0.549272 0.037019 0.4985260.057588 0.45204 0.677206 0.945677 0.300089 0.486520.282052 0.125675 0.950377 0.737419 0.110111 0.4411270.739097 0.264718 0.988922 0.797327 0.276345 0.6132820.084994 0.546434 0.126377 0.114322 0.605274 0.295480.353526 0.189032 0.710807 0.753533 0.222571 0.445894

0.25959 0.608509 0.304422 0.121006 0.061281 0.270962

Bin Frequency Bin Frequency0 0 0 0

0.1 32 0.1 00.2 28 0.2 00.3 42 0.3 80.4 26 0.4 60.5 24 0.5 180.6 32 0.6 160.7 28 0.7 80.8 38 0.8 20.9 20 0.9 2

1 30 1 0More 0 More 0

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80

1 2 3 4 5 6 7 8 9

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20