Formulae for COMP 233 Final F15 PDF

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Formulae and Probability Tables for COMP233 Final Exam 1. Basic combinations: (

)

Useful Facts and Formulae

2. The binomial theorem: (

3. DeMorgan’s laws: (⋃

)

( )

∑

) ⋂

)

(

General Probability

) ⋃

(⋂

1. Classical probability: If is an event in (finite, with equally likely outcomes), ()

then ( )

()

2. Inclusion-exclusion identity: ( 3. Conditional probability of

)

()

()

( )

given that occurred:

4. Total probability rule:

() ∑ ( ) ∑ (|)( )

5. Bayes rule:

(| )( )

(|)

(|)

⋃

∑ (| )( )

6. Independent events: and are independent if ( )

( )

()

⋃

()()

Random Variables 1. Binomial ( ):

*

/ (

+ .

2. Geometric ( ):

*

+

(

,-

)

)

3. Negative Binomial ( ):

*

+ .

4. Poisson (

*

COMP 233

): +

/ (

)

()

(

) ()

,-

()

,,Page 1 of 7

(

)

() Final Exam Formulae and Tables

5. Exponential (

* + { 6. Uniform ( ):

):

() {

* | 8. Normal Distribution ( ): 7. Memoryless property:

,-

+

( )

()

*

+

()

∫

( )

] ∑ ,-

(∑

)

()

and given pdf f(x), E(x) = ∫

, - ,( ) , - ( , -) ( ) 3. Suppose are independent RVs. Then [∑

(

() ,-

√

10. Properties of , - and ( ): 1. Given pmf(x), ( ) ∑ 2. In general,

()

+

,-

√ 9. Standard Normal Distribution *

()

,-

()

()

) ∑ ( )

11. Markov’s Inequality: If is a random variable that takes only nonnegative

values, then for any value

( ), then for any value

*

,

,-

+

12. Chebyshev’s Inequality: If is a random variable with mean and variance

*|

COMP 233

,

|

+

Page 2 of 7

Final Exam Formulae and Tables

Statistics 1. Sample Mean: Let *

+ be a sample from a population determined by a

RV . Then

∑



2. Sample Variance: Let *

a RV . Then

∑

+ be a sample from a population determined by (∑ [

)

]

3. Sample Chebyshev’s Inequality: Let  and be the sample mean and sample

standard deviation of the data set *

| |

, where

. Let

+

and let ( ) be the number of elements in the set . Then, for any ≥ 1, ( )

4. The Empirical Rule: If a data set is approximately normal with sample mean 

and sample standard deviation , then the following statements are true. a. Approximately 68 percent of the observations lie within  ; b. Approximately 95 percent of the observations lie within  . ∑

(

 )( )

5. Sample Correlation Coefficient:

√∑

) ∑

(

( )

Parameter Estimation

1. Confidence Intervals on Mean. Variance Known: A (

interval for an unknown is given by {



) confidence

} √ √ Also, for the one-sided upper and lower confidence intervals, respectively,

} {  } √ √ 2. Mean of Normal Population. Variance Unknown (Small Sample): A ( ) confidence interval for an unknown is given by {

{

COMP 233

√



Page 3 of 7

√

}

Final Exam Formulae and Tables

Also, for the one-sided upper and lower confidence intervals, respectively, { } {  3. Variance of Normal √ Population: A ( unknown is given by ( {

)

Similarly, one-sided ( ( {

)

}

(

)

(

)

} √ ) confidence interval for an }

) confidence intervals are given by {

}

Hypothesis Tests 1. Hypothesis Tests on Normal Mean, Variance Known.

When testing a two-sided hypothesis *

versus

| |+ where



, √

2. Hypothesis Tests on Normal Mean, Variance Unknown.

When testing a two-sided hypothesis *

versus

| |+ where

3. Hypothesis Tests on Normal Variance.

When testing a two-sided hypothesis + (* (

*

)

versus

,



√

+) where

Regression 1. Simple Linear Regression. Suppose are responses corresponding to the values . We wish to fit these paired values with a simple linear regression model: . The least squares estimators of and of are 

∑

∑ 

∑



Linear Congruent Generator – LCG(a, c, m, x0) (

COMP 233

)

;

Page 4 of 7

Final Exam Formulae and Tables

Table 1 Standard Normal Distribution Function: ( )

√

∫

.0 .1 .2 .3 .4 .5 .6 .7 .8 .9

.00 .5000 .5398 .5793 .6179 .6554 .6915 .7257 .7580 .7881 .8159

.01 .5040 .5438 .5832 .6217 .6591 .6950 .7291 .7611 .7910 .8186

.02 .5080 .5478 .5871 .6255 .6628 .6985 .7324 .7642 .7939 .8212

.03 .5120 .5517 .5910 .6293 .6664 .7019 .7357 .7673 .7967 .8238

.04 .5160 .5557 .5948 .6331 .6700 .7054 .7389 .7704 .7995 .8264

.05 .5199 .5596 .5987 .6368 .6736 .7088 .7422 .7734 .8023 .8289

.06 .5239 .5636 .6026 .6406 .6772 .7123 .7454 .7764 .8051 .8315

.07 .5279 .5675 .6064 .6443 .6808 .7157 .7486 .7794 .8078 .8340

.08 .09 .5319 .5359 .5714 .5753 .6103 .6141 .6480 .6517 .6844 .6879 .7190 .7224 .7517 .7549 .7823 .7852 .8106 .8133 .8365 .8389

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

.8413 .8643 .8849 .9032 .9192 .9332 .9452 .9554 .9641 .9713

.8438 .8665 .8869 .9049 .9207 .9345 .9463 .9564 .9649 .9719

.8461 .8686 .8888 .9066 .9222 .9357 .9474 .9573 .9656 .9726

.8485 .8708 .8907 .9082 .9236 .9370 .9484 .9582 .9664 .9732

.8508 .8729 .8925 .9099 .9251 .9382 .9495 .9591 .9671 .9738

.8531 .8749 .8944 .9115 .9265 .9394 .9505 .9599 .9678 .9744

.8554 .8770 .8962 .9131 .9279 .9406 .9515 .9608 .9686 .9750

.8577 .8790 .8980 .9147 .9292 .9418 .9525 .9616 .9693 .9756

.8599 .8810 .8997 .9162 .9306 .9429 .9535 .9625 .9699 .9761

.8621 .8830 .9015 .9177 .9319 .9441 .9545 .9633 .9706 .9767

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

.9772 .9821 .9861 .9893 .9918 .9938 .9953 .9965 .9974 .9981

.9778 .9826 .9864 .9896 .9920 .9940 .9955 .9966 .9975 .9982

.9783 .9830 .9868 .9898 .9922 .9941 .9956 .9967 .9976 .9982

.9788 .9834 .9871 .9901 .9925 .9943 .9957 .9968 .9977 .9983

.9793 .9838 .9875 .9904 .9927 .9945 .9959 .9969 .9977 .9984

.9798 .9842 .9878 .9906 .9929 .9946 .9960 .9970 .9978 .9984

.9803 .9846 .9881 .9909 .9931 .9948 .9961 .9971 .9979 .9985

.9808 .9850 .9884 .9911 .9932 .9949 .9962 .9972 .9979 .9985

.9812 .9854 .9887 .9913 .9934 .9951 .9963 .9973 .9980 .9986

.9817 .9857 .9890 .9916 .9936 .9952 .9964 .9974 .9981 .9986

3.0 3.1 3.2 3.3 3.4

.9987 .9990 .9993 .9995 .9997

.9987 .9991 .9993 .9995 .9997

.9987 .9991 .9994 .9995 .9997

.9988 .9991 .9994 .9996 .9997

.9988 .9992 .9994 .9996 .9997

.9989 .9992 .9994 .9996 .9997

.9989 .9992 .9994 .9996 .9997

.9989 .9992 .9995 .9996 .9997

.9990 .9993 .9995 .9996 .9997

.9990 .9993 .9995 .9997 .9998

COMP 233

Page 5 of 7

Final Exam Formulae and Tables

Table 2 Values of

 = .10 n 1 3.078 2 1.886 3 1.638 4 1.533 5 1.476 6 1.440 7 1.415 8 1.397 9 1.383 10 1.372 11 1.363 12 1.356 13 1.350 14 1.345 15 1.341 16 1.337 17 1.333 18 1.330 19 1.328 20 1.325 21 1.323 22 1.321 23 1.319 24 1.318 25 1.316 26 1.315 27 1.314 28 1.313 29 1.311  1.282 Other Probabilities: * + * + * + * +

COMP 233

 = .05

 = .025

 = .01

 = .005

6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.645

12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 1.960

31.821 6.965 4.541 3.474 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.326

63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.576

*

*

+

+

Page 6 of 7

Final Exam Formulae and Tables

Table 3 Values of

 = .995  = .99  = .975  = .95  = .05  = .025  = .01 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

.0000393 .000157 .000982 .00393 .0100 .0201 .0506 .103 .0717 .115 .216 .352 .207 .297 .484 .711 .412 .554 .831 1.145 .676 .872 1.237 1.635 .989 1.239 1.690 2.167 1.344 1.646 2.180 2.733 1.735 2.088 2.700 3.325 2.156 2.558 3.247 3.940 2.603 3.053 3.816 4.575 3.074 3.571 4.404 5.226 3.565 4.107 5.009 5.892 4.075 4.660 5.629 6.571 4.601 5.229 6.262 7.261 5.142 5.812 6.908 7.962 5.697 6.408 7.564 8.672 6.265 7.015 8.231 9.390 6.844 7.633 8.907 10.117 7.434 8.260 9.591 10.851 8.034 8.897 10.283 11.591 8.643 9.542 10.982 12.338 9.260 10.196 11.689 13.091 9.886 10.856 12.401 13.484 10.520 11.524 13.120 14.611 11.160 12.198 13.844 15.379 11.808 12.879 14.573 16.151 12.461 13.565 15.308 16.928 13.121 14.256 16.047 17.708 13.787 14.953 16.791 18.493

Other Probabilities: * COMP 233

+

3.841 5.991 7.815 9.488 11.070 12.592 14.067 15.507 16.919 18.307 19.675 21.026 22.362 23.685 24.996 26.296 27.587 28.869 30.144 31.410 32.671 33.924 35.172 36.415 37.652 38.885 40.113 41.337 42.557 43.773 *

Page 7 of 7

5.024 7.378 9.348 11.143 12.832 14.449 16.013 17.535 19.023 20.483 21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170 35.479 36.781 38.076 39.364 40.646 41.923 43.194 44.461 45.772 46.979

6.635 9.210 11.345 13.277 13.086 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 38.932 40.289 41.638 42.980 44.314 45.642 46.963 48.278 49.588 50.892

 = .005 7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 23.589 25.188 26.757 28.300 29.819 31.319 32.801 34.267 35.718 37.156 38.582 39.997 41.401 42.796 44.181 45.558 46.928 48.290 49.645 50.993 52.336 53.672

+ Final Exam Formulae and Tables...