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MAT2377 Final Exam - Formula Sheet, Exams of Mathematics

Université du QuébecMathematics

A comprehensive formula sheet for the final exam in the mat2377 course. It covers important probability distributions, including the binomial, hypergeometric, negative binomial, geometric, poisson, and exponential distributions, along with their respective formulas, mean, and variance. Additionally, the document includes key test statistics, confidence intervals, and areas under the normal curve. This formula sheet serves as a valuable reference for students preparing for the final exam in the mat2377 course, which likely covers topics in probability and statistics. The level of detail and the range of concepts covered suggest that this document would be most useful for university-level students enrolled in a statistics or probability-focused course.

Typology: Exams

2021/2022

Uploaded on 07/29/2024

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MAT2377
Final Exam - Formula Sheet
Winter 2024
•Binomial distribution with parameters nand p:
P(X=x) = n
xpx(1 −p)n−x,n
x=n!
x!(n−x)!
µ=np, σ2=np(1 −p)
•Hypergeometric distribution with parameters (N, k , n):
P(X=x) = k
xN−k
n−x
N
n, µ =nk
N
•Negative binomial distribution with success rate pand ksuccesses:
P(X=x) = x−1
k−1pk(1 −p)x−k, µ =k
p, σ2=k(1 −p)
p2
•Geometric distribution with success rate p:
P(X=x) = (1 −p)x−1p(x= 1,2,3,···), µ =1
p, σ2=1−p
p2
•Poisson distribution with time interval of length tand average number of occurrence λper
unit time:
P(X=x) = e−λt (λt)x
x!(x= 0,1,2, ...), E(X) = Var(X) = λt
•Exponential distribution with parameter β:
f(x) = 1
βe−x/β,(x≥0), µ =β , σ2=β2.
1
Formula Sheet
pf3
pf4
pf5

Partial preview of the text

Download MAT2377 Final Exam - Formula Sheet and more Exams Mathematics in PDF only on Docsity!

MAT

Final Exam - Formula Sheet Winter 2024

  • Binomial distribution with parameters n and p:

P (X = x) =

n x

px(1 − p)n−x,

n x

= (^) x!(nn −! x)!

μ = np, σ^2 = np(1 − p)

  • Hypergeometric distribution with parameters (N, k, n):

P (X = x) =

k x

N −k n−x

N

n

 , μ = nkN

  • Negative binomial distribution with success rate p and k successes:

P (X = x) =

x − 1 k − 1

pk(1 − p)x−k, μ = k p

, σ^2 = k(1^ −^ p) p^2

  • Geometric distribution with success rate p:

P (X = x) = (1 − p)x−^1 p (x = 1, 2 , 3 , · · · ), μ =^1 p, σ^2 =^1 p− 2 p

  • Poisson distribution with time interval of length t and average number of occurrence λ per unit time:

P (X = x) = e−λt^ (λt)

x x!

(x = 0, 1 , 2 , ...), E(X) = Var(X) = λt

  • Exponential distribution with parameter β:

f (x) = β^1 e−x/β^ , (x ≥ 0), μ = β, σ^2 = β^2.

Important Test Statistics Confidence Intervals

X^ ¯ − μ 0 σ/√n X¯ ± zα/ 2 √^ σ n

X^ ¯ − μ 0 s/

n X¯ ± tα/ 2 √^ s n

pp^ ˆ^ −^ p^0 p 0 q 0 /n

pˆ ± zα/ 2

r pˆˆq n

r p^ ˆ^1 −^ pˆ^2 p ˆqˆ

n 1 +

n 2

 ( ˆp^1 −^ pˆ^2 )^ ±^ zα/^2

r pˆ 1 qˆ 1 n 1 +

pˆ 2 qˆ 2 n 2

X^ ¯ 1 − X¯ 2

r σ^21 n 1 +^

σ^22 n 2

( X¯ 1 − X¯ 2 ) ± zα/ 2

r σ 12 n 1

  • σ

(^22) n 2

X^ ¯ 1 − X¯ 2

r s^2 p

n 1 +

n 2

 ( X¯^1 −^ X¯^2 )^ ±^ tα/^2

r s^2 p

n 1 +

n 2

(n − 1)S^2 σ^2

(n − 1)s^2 χ^2 α/ 2 ,^

(n − 1)s^2 χ^21 −α/ 2

b 1 − β 1 s/

Sxx^ b^1 ±^ tα/^2

√^ s Sxx

Pooled variance s^2 p = (n^1 −^ 1)s

(^21) + (n 2 − 1)s (^22) n 1 + n 2 − 2

, Pooled proportion ˆp = x^1 +^ x^2 n 1 + n 2

Linear Regression:

b 1 = S Sxy xx

, b 0 = ¯y − b 1 x,¯ s^2 = Syy n^ − −^ b 12 Sxy

Upper critical values of the T distribution for υ degrees of freedom

 - z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.
  • 0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.
  • -0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.
  • -0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.
  • -0.3 0.3821 0.3783 0.3745 0.3707 0.369 0.3632 0.3594 0.3557 0.3520 0.
  • -0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.
  • -0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.
  • -0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.283 0.
  • -0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.
  • -0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.
  • -0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.
  • -1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.
  • -1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.
  • -1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.
  • -1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.
  • -1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0722 0.0708 0.0694 0.
  • -1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.
  • -1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.
  • -1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.
  • -1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.
  • -1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.
  • -2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.
  • -2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.
  • -2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.
  • -2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.
  • -2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.
  • -2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.
  • -2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.
  • -2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.
  • -2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.
  • -2.9 0.0019 0.0018 0.0017 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.
  • -3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.
  • -3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.
  • -3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.
  • -3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.
  • -3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.
    • z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0. continue
  • 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.
  • 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.
  • 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.
  • 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.
  • 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.
  • 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.
  • 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.
  • 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.
  • 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.
  • 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.
  • 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.
  • 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.
  • 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.
  • 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.
  • 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.
  • 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.
  • 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.
  • 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.
  • 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.
  • 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.
  • 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.
  • 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.
  • 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.
  • 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.
  • 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.
  • 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.
  • 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.
  • 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.
  • 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.
  • 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.
  • 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.
  • 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.
  • 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.
  • 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.
  • 3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.
  • υ\ one tail prob t. 10 t. 05 t. 025 t. 01 t. 005 t. The shade area equal to α for t = tα
  • 1 3.078 6.314 12.706 31.821 63.657 318.
  • 2 1.886 2.920 4.303 6.965 9.925 22.
  • 3 1.638 2.353 3.182 4.541 5.841 10.
  • 4 1.533 2.132 2.776 3.747 4.604 7.
  • 5 1.476 2.015 2.571 3.365 4.032 5.
  • 6 1.440 1.943 2.447 3.143 3.707 5.
  • 7 1.415 1.895 2.365 2.998 3.499 4.
  • 8 1.397 1.860 2.306 2.896 3.355 4.
  • 9 1.383 1.833 2.262 2.821 3.250 4.
  • 10 1.372 1.812 2.228 2.764 3.169 4.
  • 11 1.363 1.796 2.201 2.718 3.106 4.
  • 12 1.356 1.782 2.179 2.681 3.055 3.
  • 13 1.350 1.771 2.160 2.650 3.012 3.
  • 14 1.345 1.761 2.145 2.624 2.977 3.
  • 15 1.341 1.753 2.131 2.602 2.947 3.
  • 16 1.337 1.746 2.120 2.583 2.921 3.
  • 17 1.333 1.740 2.110 2.567 2.898 3.
  • 18 1.330 1.734 2.101 2.552 2.878 3.
  • 19 1.328 1.729 2.093 2.539 2.861 3.
  • 20 1.325 1.725 2.086 2.528 2.845 3.
  • 21 1.323 1.721 2.080 2.518 2.831 3.
  • 22 1.321 1.717 2.074 2.508 2.819 3.
  • 23 1.319 1.714 2.069 2.500 2.807 3.
  • 24 1.318 1.711 2.064 2.492 2.797 3.
  • 25 1.316 1.708 2.060 2.485 2.787 3.
  • 26 1.315 1.706 2.056 2.479 2.779 3.
  • 27 1.314 1.703 2.052 2.473 2.771 3.
  • 28 1.313 1.701 2.048 2.467 2.763 3.
  • 29 1.311 1.699 2.045 2.462 2.756 3.
  • ∞ 1.282 1.645 1.960 2.326 2.576 3.