Chapter 3 Appendix

3.1 Common discrete distributions

Bernoulli

  • pmf: \(P(X=x \mid p)=p^x(1-p)^x\); \(x=0, 1\); \(0<p<1\)

  • mean and variance: \(E(X)=p\), \(Var(X)=p(1-p)\)

Binomial

  • pmf \(P(X=x \mid n,p)={n \choose x} p^x(1-p)^{n-x}\); \(x=0, 1, 2, \ldots, n\)
  • mean and variance: \(E(X)=np\), \(Var(X)=np(1-p)\)

Geometric

  • pmf: \(P(X=x \mid p)=(1-p)^{x-1}p,\; x=1,2,\ldots\)

  • mean and variance: \(E(X)=1/p\) and \(Var(X)=(1-p)/p^2\)

or

  • pmf: \(P(X=x \mid p)=(1-p)^xp.\)
  • mean and variance: \(E(X)=1/p-1\) and \(Var(X)=(1-p)/p^2\)

Negative Binomial

  • pmf: \(P(X=x \mid r, p)={x+r-1 \choose r-1}p^r(1-p)^x, \; x=0,1,2,\ldots\)
  • mean and variance: \(E(X)=\frac{r(1-p)}{p}\) and \(Var(X)=\frac{r(1-p)}{p^2}\).

Hypergeometric

  • pmf: \(P(X=x\mid M, N, n)=\frac{{{M \choose x}{N-M \choose n-x}}}{{N \choose n}}\); \(x=0,1,2,\ldots, n\);

    \(max\bigg(0, n-(N-M)\bigg) \le x\le min(n, M)\); \(N, M, K>0\)

  • mean and variance: \(E(X)=n\cdot \frac{M}{N}\), \(Var(X)=\left(\frac{N-n }{N-1}\right )\cdot n \cdot \frac{M}{N}\cdot (1- \frac{M}{N})\)

Poisson

  • pmf: \(P(X=x \mid \lambda)=\frac{e^{-\lambda}\lambda^x}{x!}, \;\; x=0, 1, 2, \ldots.\)

  • mean and variance: \(E(X)=Var(X)=\lambda\).

3.2 Common continuous distributions

Normal

  • pdf \(f(x\mid \mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left[-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right], \;\;\;-\infty<x<\infty\)
  • mean and variance: \(E(X)=\mu\), \(Var(X)=\sigma^2\)
  • notes: sometimes called the Gaussian distribution.

Exponential

  • pdf: \(f(x\mid \lambda)=\lambda e^{-\lambda x}\); \(x \ge 0\); \(\lambda>0\)
  • mean and variance \(E(X)=1/\lambda\); \(Var(X)=1/\lambda^2\)

Gamma

  • pdf: \(f(x\mid \alpha,\beta)=\frac{1}{\beta^\alpha\Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta}\); \(x\ge 0\); \(\alpha,\beta>0\)

  • mean and variance \(E(X)=\alpha\beta\), \(Var(X)=\alpha\beta^2\)

Chi-squared

  • pdf: \(f(x\mid \nu)=\frac{1}{2^{\nu/2}\Gamma(\nu/2)}x^{\frac{\nu}{2}-1}e^{-\frac{x}{2}}\); \(x\ge 0\); \(\nu=1, 2, 3,\ldots\)

  • mean and variance \(E(X)=\nu\), \(Var(X)=2\nu\)

Beta

  • pdf: \(f(x\mid \alpha,\beta)=\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}\)
  • mean and variance: \(E(X)=\frac{\alpha}{\alpha+\beta}\), \(Var(X)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\)

3.3 Table of standard normal distribution

Each cell represents the probability \[ \Phi(z)=P(Z\le z). \]

z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
-3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
-3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
-3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
-3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
-3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
-2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
-2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
-2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
-2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
-2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
-2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
-2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
-2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
-2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
-2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
-1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
-1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
-1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
-1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
-1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
-1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
-1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
-1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
-0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
-0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
-0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
-0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
-0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
-0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
-0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
-0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
-0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

3.3.1 Confidence Interval Critical Values, \(z_{α/2}\)

Level of Confidence, \(100(1-\alpha)\%\) Critical Value, \(z_{α/2}\)
\(90\%\) \(1.645\)
\(95\%\) \(1.96\)
\(98\%\) \(2.33\)
\(99\%\) \(2.575\)

3.3.2 Hypothesis Testing Critical Values

Level of Significance, \(\alpha\) Left-Tailed Right-Tailed Two-Tailed
\(0.10\) \(- 1.28\) \(1.28\) \(\pm 1.645\)
\(0.05\) \(- 1.645\) \(1.645\) \(\pm 1.96\)
\(0.01\) \(-2.33\) \(2.33\) \(\pm2.575\)

3.4 Table of \(t\)-critical values

Each cell represents the value of \(t_{\alpha,\nu}\).

\(t_{\alpha,\nu}\) \(\alpha=0.1\) \(\alpha=0.075\) \(\alpha=0.05\) \(\alpha=0.025\) \(\alpha=0.01\) \(\alpha=0.005\) \(\alpha=0.0005\)
\(\nu= 1\) 3.078 4.165 6.314 12.706 31.821 63.657 636.619
\(\nu=2\) 1.886 2.282 2.92 4.303 6.965 9.925 31.599
\(\nu=3\) 1.638 1.924 2.353 3.182 4.541 5.841 12.924
\(\nu=4\) 1.533 1.778 2.132 2.776 3.747 4.604 8.61
\(\nu=5\) 1.476 1.699 2.015 2.571 3.365 4.032 6.869
\(\nu=6\) 1.44 1.65 1.943 2.447 3.143 3.707 5.959
\(\nu=7\) 1.415 1.617 1.895 2.365 2.998 3.499 5.408
\(\nu=8\) 1.397 1.592 1.86 2.306 2.896 3.355 5.041
\(\nu=9\) 1.383 1.574 1.833 2.262 2.821 3.25 4.781
\(\nu=10\) 1.372 1.559 1.812 2.228 2.764 3.169 4.587
\(\nu=11\) 1.363 1.548 1.796 2.201 2.718 3.106 4.437
\(\nu=12\) 1.356 1.538 1.782 2.179 2.681 3.055 4.318
\(\nu=13\) 1.35 1.53 1.771 2.16 2.65 3.012 4.221
\(\nu=14\) 1.345 1.523 1.761 2.145 2.624 2.977 4.14
\(\nu=15\) 1.341 1.517 1.753 2.131 2.602 2.947 4.073
\(\nu=16\) 1.337 1.512 1.746 2.12 2.583 2.921 4.015
\(\nu=17\) 1.333 1.508 1.74 2.11 2.567 2.898 3.965
\(\nu=18\) 1.33 1.504 1.734 2.101 2.552 2.878 3.922
\(\nu=19\) 1.328 1.5 1.729 2.093 2.539 2.861 3.883
\(\nu=20\) 1.325 1.497 1.725 2.086 2.528 2.845 3.85
\(\nu=21\) 1.323 1.494 1.721 2.08 2.518 2.831 3.819
\(\nu=22\) 1.321 1.492 1.717 2.074 2.508 2.819 3.792
\(\nu=23\) 1.319 1.489 1.714 2.069 2.5 2.807 3.768
\(\nu=24\) 1.318 1.487 1.711 2.064 2.492 2.797 3.745
\(\nu=25\) 1.316 1.485 1.708 2.06 2.485 2.787 3.725
\(\nu=26\) 1.315 1.483 1.706 2.056 2.479 2.779 3.707
\(\nu=27\) 1.314 1.482 1.703 2.052 2.473 2.771 3.69
\(\nu=28\) 1.313 1.48 1.701 2.048 2.467 2.763 3.674
\(\nu=29\) 1.311 1.479 1.699 2.045 2.462 2.756 3.659
\(\nu=30\) 1.31 1.477 1.697 2.042 2.457 2.75 3.646
\(\nu=35\) 1.306 1.472 1.69 2.03 2.438 2.724 3.591
\(\nu=40\) 1.303 1.468 1.684 2.021 2.423 2.704 3.551
\(\nu=45\) 1.301 1.465 1.679 2.014 2.412 2.69 3.52
\(\nu=50\) 1.299 1.462 1.676 2.009 2.403 2.678 3.496
\(\nu=60\) 1.296 1.458 1.671 2 2.39 2.66 3.46
\(\nu=70\) 1.294 1.456 1.667 1.994 2.381 2.648 3.435
\(\nu=80\) 1.292 1.453 1.664 1.99 2.374 2.639 3.416
\(\nu=100\) 1.29 1.451 1.66 1.984 2.364 2.626 3.39
\(\nu=500\) 1.283 1.442 1.648 1.965 2.334 2.586 3.31
\(\nu=1000\) 1.282 1.441 1.646 1.962 2.33 2.581 3.3
\(\nu=\infty\) 1.282 1.44 1.645 1.96 2.326 2.576 3.291