Math 2.0

Learn Year 1

Probability

178 ideas in teaching order. Each idea comes after the ideas that it builds on. 174 of them have a page so far; the others are on the list in grey.

Start with Probability theory →

  1. Probability theory concept
  2. Random experiment concept
  3. Sample space definition
  4. Outcome definition
  5. Event definition
  6. Sigma-algebra definition
  7. Borel sigma-algebra definition
  8. Probability axioms concept
  9. Probability measure definition
  10. Probability space definition
  11. Probability of the complement statement
  12. Monotonicity of probability statement
  13. Addition rule for two events statement
  14. Boole's inequality statement
  15. Continuity of probability measures statement
  16. Classical definition of probability definition
  17. Dirac measure definition
  18. Geometric probability concept
  19. Buffon's needle problem problem
  20. Bertrand paradox example
  21. Conditional probability definition
  22. Multiplication rule statement
  23. Law of total probability statement
  24. Law of total probability for an event and its complement statement
  25. Bayes' theorem statement
  26. Prior probability concept
  27. Posterior probability concept
  28. Base rate fallacy example
  29. Monty Hall problem problem
  30. Independence of two events definition
  31. Mutual independence of events definition
  32. Pairwise independence definition
  33. Pairwise independent events that are not mutually independent counterexample
  34. Conditional independence definition
  35. Independent sigma-algebras definition
  36. Bernoulli trial concept
  37. Birthday problem example
  38. Random variable definition
  39. indicator random variable definition
  40. Probability distribution definition
  41. Law of a random variable definition
  42. Identically distributed random variables definition
  43. Cumulative distribution function definition
  44. Monotonicity of the cumulative distribution function statement
  45. Limit of the cumulative distribution function at infinity statement
  46. The cumulative distribution function determines the distribution statement
  47. Discrete random variable definition
  48. Probability mass function definition
  49. Random variable with a density definition
  50. Probability density function definition
  51. Quantile function definition
  52. median definition
  53. Discrete uniform distribution definition
  54. Bernoulli distribution definition
  55. Binomial distribution definition
  56. Geometric distribution definition
  57. Negative binomial distribution definition
  58. Hypergeometric distribution definition
  59. Poisson distribution definition
  60. Continuous uniform distribution definition
  61. Exponential distribution definition
  62. Memorylessness concept
  63. Normal distribution definition
  64. Density of the normal distribution definition
  65. Standard score definition
  66. Gamma distribution definition
  67. Beta distribution definition
  68. Cauchy distribution definition
  69. Joint probability distribution definition
  70. Marginal distribution definition
  71. Independence of two random variables definition
  72. Independence of random variables definition
  73. Independence means the joint law is the product of the marginals statement
  74. Independent and identically distributed random variables definition
  75. Conditional probability distribution definition
  76. Multivariate normal distribution definition
  77. Expected value definition
  78. Law of the unconscious statistician statement
  79. Linearity of expectation statement
  80. Expectation of a product of independent random variables statement
  81. Tail sum formula for expectation statement
  82. Expectation with respect to the binomial distribution statement
  83. Mean of the Poisson distribution statement
  84. Mean of the geometric distribution statement
  85. Mean of the normal distribution statement
  86. Variance definition
  87. Variance as the mean of the square minus the square of the mean statement
  88. Variance under scaling statement
  89. Variance under translation statement
  90. Variance of the normal distribution statement
  91. Standard deviation definition
  92. Covariance definition
  93. Variance of a sum statement
  94. Variance of a sum of independent random variables statement
  95. Independent random variables are uncorrelated statement
  96. Uncorrelated random variables that are not independent counterexample
  97. Correlation coefficient definition
  98. Covariance matrix definition
  99. Moment definition
  100. Central moment definition
  101. Skewness definition
  102. Kurtosis definition
  103. Probability-generating function definition
  104. Generating function of a sum of independent random variables statement
  105. Moment-generating function definition
  106. Moments are derivatives of the moment-generating function at zero statement
  107. Moment-generating function of a sum of independent random variables statement
  108. Moment-generating function of the normal distribution statement
  109. Cumulant generating function definition
  110. characteristic function definition
  111. Characteristic function of a sum of independent random variables statement
  112. Characteristic function of the normal distribution statement
  113. The characteristic function determines the distribution statement
  114. Markov's inequality statement
  115. Chebyshev's inequality statement
  116. Chernoff bound statement
  117. Cauchy–Schwarz inequality for expectations statement
  118. Hoeffding's inequality statement
  119. Conditional expectation definition
  120. Law of total expectation statement
  121. Conditional variance definition
  122. Law of total variance statement
  123. Convolution of probability distributions definition
  124. Sum of independent Poisson random variables statement
  125. Sum of independent normal random variables statement
  126. Convergence of random variables concept
  127. Almost surely definition
  128. Almost sure convergence definition
  129. Convergence in probability definition
  130. Convergence in distribution definition
  131. Convergence in mean definition
  132. Almost sure convergence implies convergence in probability statement
  133. Convergence in probability implies convergence in distribution statement
  134. First Borel–Cantelli lemma statement
  135. Second Borel–Cantelli lemma statement
  136. Kolmogorov's zero–one law statement
  137. Law of large numbers concept
  138. Weak law of large numbers statement
  139. Strong law of large numbers statement
  140. Poisson limit theorem statement
  141. Central limit theorem statement
  142. De Moivre–Laplace theorem statement
  143. Lévy's continuity theorem statement
  144. 68–95–99.7 rule example
  145. Stochastic process concept
  146. Bernoulli process definition
  147. Random walk definition
  148. Gambler's ruin problem
  149. Markov chain definition
  150. Markov property definition
  151. transition matrix definition
  152. Chapman–Kolmogorov equation statement
  153. Stationary distribution definition
  154. Absorbing Markov chain definition
  155. Poisson point process definition
  156. Galton–Watson process definition
  157. Martingale definition
  158. Coupon collector's problem problem
  159. Statistic definition
  160. Sample mean definition
  161. Sample variance definition
  162. Estimator definition
  163. Bias of an estimator definition
  164. Mean squared error definition
  165. Consistent estimator definition
  166. Method of moments concept
  167. Likelihood function definition
  168. Maximum likelihood estimation concept
  169. Confidence interval definition
  170. Chi-squared distribution definition
  171. Student's t-distribution definition
  172. Statistical hypothesis test concept
  173. P-value definition
  174. Type I and type II errors definition
  175. Student's t-test concept
  176. Chi-squared test concept
  177. Neyman–Pearson lemma statement
  178. Bayesian inference concept