Probability theory and stochastic processes Definition
Discrete random variable
Builds on Random variableCountable set
Definition
Let be a real random variable on a probability space . is discrete if there is a countable set with . A distribution on is discrete if for some countable set .
Remarks
Then for each Borel set . So the probability mass function determines the law of . The cumulative distribution function of a discrete random variable is a step function when the set of values has no limit points. It jumps by at each value . The Bernoulli, binomial, geometric and Poisson distributions are discrete.
Reading path
To understand this page from the ground up, read these 41 pages in order. The foundations come first.
- Construction of the reals from Cauchy sequences concept
- Element definition
- Integers object
- Ordered field definition
- Proposition concept
- Random experiment concept
- Rational numbers object
- Real numbers object
- Set definition
- Ring of integers statement
- Disjunction definition
- Implication definition
- Predicate definition
- Ordered pair definition
- Union definition
- Sample space definition
- The rationals form a field statement
- Existential quantifier definition
- Universal quantifier definition
- Cartesian product definition
- Subset definition
- The rationals form an ordered field statement
- Uniqueness quantifier definition
- Binary relation definition
- Universal set definition
- Order on the reals definition
- Power set definition
- Complement definition
- Function definition
- Interval definition
- Injective function definition
- Preimage of a set definition
- Sigma-algebra definition
- Countable set definition
- Event definition
- Borel sigma-algebra definition
- Probability axioms concept
- Probability measure definition
- Probability space definition
- Random variable definition
- Discrete random variable definition
Where this leads
208 pages build on this one.
- Variance
- Chernoff bound
- Chapman–Kolmogorov equation
- Bias of an estimator
- Law of the unconscious statistician
- Covariance matrix
- Chebyshev's inequality
- Characteristic function of a sum of independent random variables
- Stationary distribution
- Bernoulli process
- Markov chain
- Law of total variance
- Variance of the normal distribution
- Cauchy–Schwarz inequality for expectations
- Correlation coefficient
- Variance under translation
- Law of total expectation
- Expectation of a product of independent random variables
- Likelihood function
- Markov's inequality
- Poisson point process
- De Moivre–Laplace theorem
- Independent random variables are uncorrelated
- Convolution of probability distributions
- Mean of the Poisson distribution
- Method of moments
- Mean of the normal distribution
- Moment-generating function of the normal distribution
- Weak law of large numbers
- Uncorrelated random variables that are not independent
- Kurtosis
- Martingale
- Variance of a sum
- Mean of the geometric distribution
- Expected value
- Strong law of large numbers
- Covariance
- Neyman–Pearson lemma
- Chi-squared test
- Central limit theorem
- 68–95–99.7 rule
- Transition matrix
- Conditional variance
- Lévy's continuity theorem
- Convergence of random variables
- Maximum likelihood estimation
- Skewness
- Galton–Watson process
- Markov property
- Student's t-test
- Bernoulli distribution
- Sample variance
- Bayesian inference
- Confidence interval
- Variance under scaling
- Discrete uniform distribution
- Coupon collector's problem
- Characteristic function
- Variance as the mean of the square minus the square of the mean
- Moment-generating function
- Student's t-distribution
- Characteristic function of the normal distribution
- Sum of independent normal random variables
- Linearity of expectation
- Moment
- Expectation with respect to the binomial distribution
- Cumulant generating function
- Sample mean
- Generating function of a sum of independent random variables
- Chi-squared distribution
- Hypergeometric distribution
- Moments are derivatives of the moment-generating function at zero
- The characteristic function determines the distribution
- Convergence in mean
- Variance of a sum of independent random variables
- Mean squared error
- Poisson distribution
- Binomial distribution
- Standard deviation
- Probability mass function
- Hoeffding's inequality
- Moment-generating function of a sum of independent random variables
- Geometric distribution
- Poisson limit theorem
- Negative binomial distribution
- Tail sum formula for expectation
- Conditional expectation
- Absorbing Markov chain
- Sum of independent Poisson random variables
- Conditional probability distribution
- Probability-generating function
- Memorylessness
- Central moment
- Doob's upcrossing inequality
- Cramér–Rao bound
- Efficiency of an estimator
- Poisson regression
- Ergodic decomposition
- Koksma–Hlawka inequality
- Hodges' superefficient estimator
- Martingale representation theorem
- Cramér–Wold theorem
- Boolean model
- Wald–Wolfowitz optimality of the sequential probability ratio test
- Doob's martingale inequality
- Consistency of the maximum likelihood estimator
- Itô isometry
- Bayes factor
- Basu's theorem
- Lindley's paradox
- M-estimator
- Score test
- Kingman's subadditive ergodic theorem
- McDiarmid's inequality
- Lévy's characterization of Brownian motion
- Asymptotic normality of the maximum likelihood estimator
- Score
- F-distribution
- Edgeworth expansion
- Monotone likelihood ratio
- Sample mean and sample covariance
- Wilcoxon signed-rank test
- Metropolis–Hastings algorithm
- Continuous-time Markov chain
- Minimal sufficient statistic
- Whittle likelihood
- Logistic regression
- Convergence to equilibrium of ergodic Markov chains
- Azuma's inequality
- Gaussian free field
- Erdős–Kac theorem
- Detailed balance
- Coupling from the past
- Hidden Markov model
- Continuous mapping theorem
- Bias–variance decomposition
- Wald's equation
- Monte Carlo integration
- Optional stopping theorem
- Breakdown point
- Delta method
- Noncentral chi-squared distribution
- Hájek–Le Cam convolution theorem
- Behrens–Fisher problem
- Submartingale
- Kruskal–Wallis test
- Information geometry
- Recurrence and transience of states
- Malliavin calculus
- Wishart distribution
- Birth–death process
- Minimum-variance unbiased estimator
- Sign test
- Slutsky's theorem
- Tanaka's formula
- Empirical likelihood
- Bayesian information criterion
- Berry–Esseen theorem
- Cochran's theorem
- Stochastic approximation
- Jackknife
- Markov chain mixing time
- Birkhoff's ergodic theorem
- Akaike information criterion
- Credible interval
- Independence of sample mean and sample variance for normal samples
- Gibbs sampling
- Estimating equations
- Supermartingale
- Spearman's rank correlation coefficient
- Markov chain Monte Carlo
- Generalized method of moments
- Doob's martingale convergence theorems
- Bernstein inequalities
- Semimartingale
- Bootstrap
- Influence function
- Oseledets theorem
- Itô calculus
- Complete statistic
- Horvitz–Thompson estimator
- U-statistic
- Sufficient statistic
- Missing data mechanisms
- Likelihood-ratio test
- Conjugate prior
- Expectation–maximization algorithm
- Hotelling's T-squared distribution
- Burkholder–Davis–Gundy inequality
- Doob–Meyer decomposition theorem
- Wald test
- Generalized linear model
- Box–Jenkins method
- Pesin theory
- Binomial proportion confidence interval
- Lévy's upward theorem
- Neyman construction
- Wilks' theorem
- Skorokhod's embedding theorem
- Fisher's exact test
- Doob decomposition theorem
- Frequentist inference
- Likelihood principle
- Fisher–Neyman factorization theorem
- Girsanov theorem
- Observed information
- Point estimation
- Local martingale
Further reading
- Web https://en.wikipedia.org/wiki/Probability_distribution, Discrete probability distribution
About this page
| Tag | 0J6Z6H5VQJ |
|---|---|
| URI | m2://0J6Z6H5VQJ |
| Kind | definition |
| Writing | W0 stub |
| MSC | 60E05 |
| Defines | Discrete random variable |
| Uses | Random variable, Countable set |
Markdown · JSON · Edit on GitHub
What this page needs for W1
- an inventory row: the row of the inventory that the node accounts for (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.