Ordered pair

What depends on this

379 nodes use this node, directly or through other nodes.

• Closure is the set of limits of sequences 003QP7A8HG
• Variance 007QED3E0Y
• Finite intersection property 00BWV5CKCF
• Hasse diagram 00EYM7MDKS
• Compact set 00R0X1WGPX
• Second-countable space 00RE1R4W7N
• Partial order 011C29Q1S1
• Chernoff bound 014PEBGDYF
• Chapman–Kolmogorov equation 01APZ8GDQH
• Bias of an estimator 01BD2TFFP4
• Totally disconnected space 01CYW8S04D
• Urysohn's lemma 01EY3VR7ZD
• Composition of injections is injective 01FPXMTBX1
• Order of cardinals 01GTP8M5RG
• Equivalence classes form a partition 01HT2CH8BH
• Triangle inequality for a metric 01M6CW9VM6
• Antisymmetric relation 01MGYYBB9R
• Infinite set 01MKW4R283
• Isolated point 01PKQYTTMT
• Composition of relations 01SWBKDAWA
• Bijective iff it has a two-sided inverse 01VG2A0BHW
• Cantor set is closed 01VT24J1SD
• Identically distributed random variables 01X8YSDTBW
• Open set 01ZQQDCERP
• Law of the unconscious statistician 01ZWX875AT
• Well-founded relation 0230ECX1Q5
• Left inverse 02486SD328
• Density of the normal distribution 027JF1FBB0
• Exponential distribution 028Z8WC5Y9
• Covariance matrix 02CBHV4ZZQ
• Open balls form a basis of the metric topology 02EEX96DFS
• Chebyshev's inequality 02HZ1BK3MW
• Isometries are injective 02J4G4J4TJ
• Characteristic function of a sum of independent random variables 02JGYG0T7M
• Stationary distribution 02KYGSCTCN
• Closure is closed 02TZDKG50Q
• Bernoulli process 02XPXR1FBK
• Total order 0316AYH1SC
• Markov chain 033TJBZDG0
• Statistic 036756NFD6
• Arzelà–Ascoli theorem 0381S8Z5P0
• Law of total variance 03BY5XTNA4
• Variance of the normal distribution 03EESZDB9A
• Equivalence of norms in finite dimension 03MC70ZRNR
• Torus 03NM7DZXXC
• Cauchy–Schwarz inequality for expectations 03PGHKAJG3
• Lebesgue number lemma 040CFNJEAP
• Continuous bijection from a compact space to a Hausdorff space is a homeomorphism 0430ETCMKD
• Product metric 044D4AQV8V
• Möbius strip 04AJE0M271
• Correlation coefficient 04CTY6XYWN
• Variance under translation 04CYN5G6RY
• Almost sure convergence 04P7D63W8P
• Law of total expectation 04RRB6HM00
• Zorn's lemma 04RXTQA5ZA
• Product topology on an arbitrary product 04S9HX3GAP
• Probability density function 04YG690N5V
• The real numbers are uncountable 0518MM6BE1
• Compact iff complete and totally bounded 054Q5PW5GX
• Expectation of a product of independent random variables 05AW1G0WE1
• Closed ball 05DSV40EB9
• Likelihood function 05FSTXM92W
• T1 space 05GQJXBP3F
• Homeomorphism 05GZ11PWRY
• Composition of continuous maps is continuous 05NK26G0DR
• Markov's inequality 05PWWHYBX5
• Standard score 05RZ4YZG4E
• Poisson point process 05TG0SE4HT
• Gamma distribution 06395GBNWD
• Metric space 063W5JMY85
• Inclusion of a subspace is continuous 065J822CKN
• De Moivre–Laplace theorem 065X9S8CM3
• Closed subsets of compact sets are compact 066HW6D9A7
• The integers are countable 066RJJVWH9
• Closure in a metric space 068P0P32DW
• Diameter of a set 0692CGQF00
• Heine–Cantor theorem 06B8BNJCD6
• Convergence of a sequence in a metric space 06JYPJ4WHH
• Kolmogorov's zero–one law 073Q8Z94KG
• Correspondence between equivalence relations and partitions 074482Q7CD
• The reals have the cardinality of the continuum 075PM9FH7A
• Independent random variables are uncorrelated 077GN9HDR6
• Convolution of probability distributions 07CHJ5YX28
• Lipschitz maps are uniformly continuous 07DCKDXRBP
• Composition of surjections is surjective 07EH4R3RVQ
• Function composition 07HF9YCWPP
• Continuity at a point of a topological space 07N7W5C29X
• Heine–Borel theorem 07PFYV65MB
• Normal distribution 07RXBVTVZB
• Distance from a point to a set 07TXYYVYZR
• Absolutely convergent series in a Banach space converge 082MGNWNA8
• Equivalence class 082RMRSY26
• Mean of the Poisson distribution 086HEVXR9Z
• Isometry 0878DN3WCS
• Statistical hypothesis test 087P03NEND
• Binary relation 08EJC6Q25H
• Equinumerous sets 08EJHHMB5R
• Complete metric space 08EWT7J0SC
• Product of countable sets is countable 08G4XH86JC
• Sequence spaces lp 08P6T2WH6R
• Independence means the joint law is the product of the marginals 08QCA2F1NX
• Topological embedding 08SVRVHJNY
• Median 08W2CW8PKW
• Domain 093CM69AGS
• Riesz's lemma 097C7KCAT5
• Intermediate value theorem for connected spaces 098C0KZZAP
• Dirac measure 099SXAG014
• Open sets characterised by neighbourhoods 09CJ00H3NW
• Closed sets contain their limits 09GQ3ZTQJ9
• Completeness of the space of bounded continuous functions 09GRCZVR5E
• Method of moments 09J1H07VAH
• Metrizable space 09KDHB82SG
• Finite intersection of open sets is open 09NDE9V7X1
• Mean of the normal distribution 09PR1V6RWR
• Intersection of closed sets is closed 09YMV8H4X5
• Product of compact sets is compact 09ZKE5XQ74
• Open sets of a metric space 0A26X79RTB
• Metric topology 0A6YRNZB2R
• Baire category theorem 0A7KGAYHGV
• Schröder–Bernstein theorem 0A9V6RKQG6
• Finite-dimensional normed spaces are complete 0ABP2F2ZP7
• Metrizable spaces are normal 0AD0BZ49DW
• Normal space 0AJ646N339
• Independent and identically distributed random variables 0AJN7WHBJZ
• Classical definition of probability 0AKNPYQ04R
• Marginal distribution 0AXP2Z5N9H
• Moment-generating function of the normal distribution 0B1G4E21G7
• Open balls are open 0B2305GNAZ
• Quantile function 0B3Q3XDM5X
• Continuity via closed sets 0B8GPXFSP7
• Connected subsets of the real line are intervals 0B9ZQFMQNV
• Finite set 0BBTH3YSZA
• Banach space 0BDTTM24DP
• Weak law of large numbers 0BFDHF2H7N
• Random walk 0C08YMV95H
• Linear maps on finite-dimensional spaces are continuous 0C13E623WD
• Random variable with a density 0C21RZG5V2
• Interior 0C38S0KMJK
• Uncorrelated random variables that are not independent 0C6J8BQXVR
• Cartesian product 0C81YBVTWH
• Kurtosis 0CBQVVNBMP
• Martingale 0CDVK9RQBH
• Box topology 0CTM1EHG9P
• Transitive relation 0CYRECG2YQ
• Dense set 0D198GPMTJ
• Restriction of a function 0D2MXBV21T
• A set is closed iff it equals its closure 0D2W7FS52T
• Baire space 0D5KR33H7X
• Variance of a sum 0D86GJEZ39
• Limits are unique in Hausdorff spaces 0D8T984XEE
• Connected space 0D939CMME7
• Mean of the geometric distribution 0DB0R4N5XX
• Open ball 0DC3MPR7C9
• Expected value 0DCT0QZVQ1
• Right inverse 0DDPZAG5ER
• Strong law of large numbers 0DFWHQ839T
• Covariance 0DJ8W3HP2D
• Random variable 0DKRR22AEQ
• Closure 0DRKT1QJR7
• The rationals are countable 0DS9KWFDK2
• Independence of two random variables 0E9502NT92
• Range of a function 0E9GTAV478
• Neyman–Pearson lemma 0EE19CFEEF
• Convergence in probability 0EFDDXTVD6
• P-value 0EHYZFZ4PK
• Epsilon-delta continuity in metric spaces 0EKGKY5TGH
• Chi-squared test 0EN0V3NY20
• Continuous image of a connected set is connected 0EPD87YVG8
• Convergence in probability implies convergence in distribution 0EPTDZPJ83
• Central limit theorem 0EPZC4Z1FC
• Locally compact space 0EQMWQ4WMJ
• 68–95–99.7 rule 0EYQ03XCR3
• Cantor's intersection theorem 0EZBG8SKJM
• Well-order 0F1D6PK9RZ
• Transition matrix 0F1KBVY161
• Conditional variance 0F1SN93KXE
• Bounded linear maps are continuous 0F3K4C7XYW
• Codomain 0F9JHP8AJN
• Closed subsets of complete spaces are complete 0FBXJZX3MK
• Universal property of the quotient topology 0FF2FJ8CH2
• Preimage of a set 0FGCFY1AZX
• Epsilon-delta uniform continuity 0FHSDCQACY
• Lévy's continuity theorem 0FPWF7TTMB
• Convergence of random variables 0FQ968YKA1
• Meagre set 0FQPYAY9PJ
• Maximum likelihood estimation 0FREPFKNS7
• Identity function 0FSNQ0GNCC
• Completion of a metric space 0G4J7Z7WPJ
• Multivariate normal distribution 0G558GJQMW
• Closed set 0G5AV9Q4RR
• Lexicographic order 0G77XFBV00
• Separable space 0GKJ78948V
• Antichain 0GNGPF7KS5
• Chain 0GP9ASVQ1X
• Cardinality of the power set 0GPFKS96T7
• Skewness 0GQ3H7VW3V
• Pseudometric space 0H3W614GEC
• Upper bound 0H5J02P8S9
• Galton–Watson process 0H631CZP5K
• Interior is the largest open subset 0HAZEFH137
• Markov property 0HB8SXHPFA
• Limit point 0HBQC2GJXT
• Union of open sets is open 0HEF46RPEK
• First-countable space 0HG11FW8MJ
• Axiom of choice 0HTGB13FE2
• Injective iff it has a left inverse 0HYFEEZAEM
• Student's t-test 0HYP606RYF
• Equivalent norms 0J3578562S
• Associativity of composition 0J6AKYZQT8
• Discrete random variable 0J6Z6H5VQJ
• Bernoulli distribution 0J97P5KZY0
• Converse relation 0JCSQNNP2M
• Convergence in distribution 0JDAQHWA9N
• The cumulative distribution function determines the distribution 0JEVZAFDY6
• Regular space 0JF9KW2TXB
• Boundary 0JJ9J1XZ8G
• Sample variance 0JN9P4A944
• Cardinality 0JTGRV4WT4
• Tychonoff's theorem 0JTRSJKCFB
• Bayesian inference 0JWRJX2RMP
• Path-connected space 0K0GE6BCYW
• Topologist's sine curve 0K3T4A659R
• Confidence interval 0K4NMMJBG2
• Complement of an open set is closed 0K67Q60Z8P
• Compactness and sequential compactness agree in metric spaces 0K8SGQSV7G
• Continuous image of a compact set is compact 0K9AAJ8G51
• Hausdorff maximal principle 0KM4KVR9MV
• p-norm on R^n 0KNJWEVSMG
• Nowhere dense set 0KSM23SN3H
• Variance under scaling 0KVQ9D86BM
• Discrete uniform distribution 0KYP951QWH
• Universal property of the product topology 0M6W4XNST6
• Epsilon characterisation of Cauchy sequences 0MCK8K2D7Z
• Minimal element 0MCR8ZAXQD
• One-point compactification 0MM45BT6HC
• Continuum hypothesis 0MN2MER9WN
• Compact Hausdorff spaces are normal 0MQQQ5NA2S
• Contraction mapping 0MT2S0ZX4Y
• Equicontinuity 0MW2WXTHMC
• Hausdorff space 0MXB4CKHFW
• Coupon collector's problem 0MXCP7MXWJ
• Cumulative distribution function 0MXMS3AE13
• Open connected subsets of R^n are path-connected 0MXN7AFBPE
• Characteristic function 0MYDNRT7PP
• Compact subsets of a Hausdorff space are closed 0MYFJVMN7M
• Well-ordering theorem 0N109NG5ZX
• Completeness via Cauchy sequences 0N1ZWFJDN5
• Hilbert's hotel 0N3Y8G340X
• Symmetric relation 0N9SA25Z3V
• Cauchy distribution 0NA6Q82YR7
• Variance as the mean of the square minus the square of the mean 0NMF95YK8W
• Epsilon-net characterisation of total boundedness 0NMGJJ19HD
• Birthday problem 0NMWG2PHJ0
• Moment-generating function 0NNRNGKMFM
• Cantor pairing function 0NP5N5PYW5
• Student's t-distribution 0NQ5798SB1
• Connected components are closed 0NRT0E2QT2
• Characteristic function of the normal distribution 0NS0TDA4E2
• Open cover 0NVPNC12JY
• Sum of independent normal random variables 0NXGPJWNFN
• Closed map 0NYZ2M45G9
• Compactness via open covers 0P6PMJB4MA
• Linearity of expectation 0P7PTC8DDE
• Bounded set 0P953MG05E
• Metric spaces are Hausdorff 0P9DCDES73
• Sup norm on bounded continuous functions 0PARCKE8WP
• Transitive closure 0PEJH7AC74
• Moment 0PH0VHTQS0
• Heine–Borel theorem in finite-dimensional normed spaces 0PHT6NBYPB
• Projections are continuous 0PM88Z9W4M
• Interior is open 0PNWCM559P
• Beta distribution 0PP5ZAJPP7
• Expectation with respect to the binomial distribution 0PVCTCKD3C
• Injective function 0PVG1CRAJN
• Cumulant generating function 0Q2QC81VTW
• Image of a set 0Q3F20XB0A
• Joint probability distribution 0QAR0XDZH4
• Equivalence relation induced by a function 0QE5PFJY7V
• Stochastic process 0QH5ENKZM1
• Cardinal number 0QKPNZKNEA
• Sample mean 0QNRBHWV35
• Compact space 0QPHW703PV
• Connected set 0R37DFJKFW
• T0 space 0R3B7DX55J
• Type I and type II errors 0R41PVFA39
• Compact subsets of a metric space are bounded 0R7BTEWCJ2
• Ordinal number 0RDJK7EQYY
• Generating function of a sum of independent random variables 0REMT01N36
• Chi-squared distribution 0RFMMBHFTV
• Limit of the cumulative distribution function at infinity 0RJ5TZH6G0
• Cantor set is homeomorphic to the space of binary sequences 0RQ8DHHR8V
• Every infinite set has a countably infinite subset 0RWQ5VZ6VQ
• Image of an intersection 0RX7DDBYCK
• Preorder 0RXS8R6K37
• Continuous linear map 0S3SMRH0Z5
• Path component 0SB9FNZECG
• Countably infinite set 0SEZ3JGZP7
• Equivalence relation 0SGTMBEWRF
• Closure is the smallest closed superset 0SHC6Z304Z
• Bijective function 0SNQEF7Z53
• Hypergeometric distribution 0SPBPW3K61
• Open map 0SQ2PC596Y
• Moments are derivatives of the moment-generating function at zero 0STK8KEY9G
• Almost sure convergence implies convergence in probability 0SVN6X695K
• Continuous map of topological spaces 0SY9ECQ19Q
• Bertrand paradox 0T0JWJ3M47
• Least element 0T2NTJCV83
• Subspace metric 0T2WYDTRP2
• The characteristic function determines the distribution 0T8BYAFEXX
• Lipschitz map 0TMZ21WKCN
• Law of a random variable 0TST4MHA3A
• Convergence in mean 0TTA57SEJG
• Estimator 0TTNRP5PFQ
• Equivalent metrics 0TWQ5PRKWS
• Surjective iff it has a right inverse 0TYQNHVB4C
• Maximal element 0TZEEFTNCC
• Variance of a sum of independent random variables 0V0778DCW9
• Aleph-null 0V2H5PGJ9B
• Continuous real function on a compact set attains its bounds 0V4DET59NW
• Mean squared error 0V4VSCBXP3
• Quotient topology 0V8046R0P9
• Poisson distribution 0VEFSSKX18
• Continuity is equivalent to sequential continuity in metric spaces 0VGPKZ8MRK
• Banach fixed point theorem 0VK6VEH345
• Isometric isomorphism 0VRD8GC5BM
• Binomial distribution 0VRWZDWW6B
• Indicator function 0VSB0XF4G7
• Standard deviation 0VTN5RKNC5
• Compact closed unit ball implies finite dimension 0VZVV8VME5
• Complete subspaces are closed 0W41XG4W60
• Inverse function 0W766JAPSN
• Cardinality of the continuum 0WC28W9DNM
• Totally bounded set 0WE5BBHGM6
• Probability mass function 0WF4EJP91W
• Monotone function 0WFF6AZPBP
• Gambler's ruin 0WFMKK7DFZ
• Continuous uniform distribution 0WGDPYR30Q
• Circle as a quotient of the real line 0WH1GVBJNH
• Hoeffding's inequality 0WPDNXGSSN
• Moment-generating function of a sum of independent random variables 0WRZ170YEH
• Cantor's theorem 0WTEMGYV2C
• Hölder continuity 0WZTM0EFR6
• Countable intersection of dense open sets is dense 0X2T63GFQQ
• Tietze extension theorem 0X52TNMTCJ
• Uncountable set 0X55M9XET4
• Independence of random variables 0XCE070V8B
• Cantor's diagonal argument 0XCH8QVDCQ
• Geometric distribution 0XCVV5Q500
• Monotonicity of the cumulative distribution function 0XD22PBMWA
• Reflexive relation 0XEGNW851S
• Function 0XHDTE5YMZ
• Consistent estimator 0XJVF81QF6
• Surjective function 0XKH881BZ7
• Poisson limit theorem 0XKTZRWTXX
• Negative binomial distribution 0XN7KEVSSS
• Graph of a function 0Y0KFHF47G
• Tail sum formula for expectation 0Y6QNF1J49
• Order isomorphism 0Y74KN99A5
• Conditional expectation 0YAMJRK04E
• Absorbing Markov chain 0YB93GWTKK
• Comparability of cardinals 0YBHSJMN3J
• Continuity via open sets 0YF82GBTND
• Sum of independent Poisson random variables 0YH1M56SBN
• Closed balls are closed 0YS40B38JD
• Conditional probability distribution 0YVDC5PYDW
• Choice function 0YXEQWRGP0
• Proper metric space 0Z4V98VN5E
• Product topology 0Z57EC6YX3
• Greatest element 0Z7HDDXX3X
• Countable union of countable sets is countable 0Z8FM54P86
• Probability-generating function 0Z8PSSAVBA
• Memorylessness 0ZAZZ34WQS
• Countable set 0ZG61TFBES
• Transfinite induction 0ZHQMQ2C1N
• Path-connected spaces are connected 0ZMT1GP4KM
• Central moment 0ZT24EC90K
• Operator norm 0ZTEH055YF
• Law of large numbers 0ZVEAYVRC3
• Quotient set 0ZVHXDSZC7