GATE ST flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Law of total probability

For a partition of the sample space into events A_i, P(B) equals the sum over i of P(B|A_i) times P(A_i).

Bayes' theorem

The posterior probability P(A_j|B) equals P(B|A_j)P(A_j) divided by the sum over i of P(B|A_i)P(A_i), the total probability of B.

Geometric distribution mean

For the number of trials until the first success with success probability p, the mean is 1/p and the variance is (1−p)/p²; the law is memoryless.

Exponential memorylessness

For X exponential with rate λ, P(X>s+t | X>s) equals P(X>t); the exponential is the only continuous distribution with this property.

Variance decomposition identity

Var(X) equals E[X²] minus the square of E[X]; also Var(aX+b) equals a²Var(X), and adding a constant never changes variance.

Zero covariance and independence

Independence implies zero covariance, but zero covariance does not imply independence in general; the converse holds for jointly normal variables.

Moment generating function

M_X(t) equals E[e^{tX}]. When finite on an open interval around zero, it uniquely determines the distribution and its derivatives at zero give the moments.

Tower property

The law of iterated expectation states E[X] equals E[E[X|Y]]; conditional expectation averages out the conditioning variable over its distribution.

Jensen's inequality

For a convex function g, E[g(X)] is at least g(E[X]); for strictly convex g, equality holds only when X is constant almost surely.

Chi-square distribution definition

If Z_1 through Z_k are independent standard normals, their squared sum is chi-square with k degrees of freedom; its mean is k and variance is 2k.

Student t statistic

A t variable with ν degrees of freedom is Z divided by the square root of V/ν, where Z is standard normal and V is independent chi-square with ν degrees of freedom.

Sample variance distribution

For a normal sample, (n−1)S²/σ² is chi-square with n−1 degrees of freedom, and the sample mean is independent of the sample variance.

F distribution ratio

The ratio of two independent chi-square variables each divided by its degrees of freedom follows an F distribution; reciprocating swaps the degrees of freedom.

Cramér–Rao lower bound

For unbiased estimators, variance is at least the reciprocal of n times Fisher information; information equals the variance of the score and minus expected second derivative of log-likelihood.

Fisher–Neyman factorization

A statistic T is sufficient for a parameter exactly when the joint density factors into a function of T and parameter times a function of the data alone.

Lehmann–Scheffé theorem

If T is a complete sufficient statistic and an estimator equals g(T) with expectation equal to the parameter, then g(T) is the unique minimum-variance unbiased estimator.

Rao–Blackwell theorem

Conditioning any unbiased estimator on a sufficient statistic yields a new unbiased estimator whose variance is no larger than that of the original estimator.

Maximum likelihood invariance

If a parameter estimate maximizes the likelihood, then the transformed estimate maximizes the induced likelihood; under regularity the MLE is asymptotically normal and efficient.

Pivot for confidence intervals

A pivot is a function of data and parameter whose distribution does not depend on the parameter; inverting probability statements about it yields exact confidence intervals.

Neyman–Pearson lemma

For simple null and alternative hypotheses, the most powerful test rejects for large values of the likelihood ratio, with the critical value chosen to achieve the size.

Duality of tests and intervals

A parameter value lies inside a 100(1−α)% confidence set exactly when the level-α test of that value as null hypothesis fails to reject.

Power of a test

Power equals one minus the type II error rate, the probability of rejecting the null when the alternative is true; it rises with effect size, sample size, and level.

Gauss–Markov theorem

Under zero-mean, constant-variance, uncorrelated errors, ordinary least squares gives the best linear unbiased estimator of the regression coefficients; normality is not required.

Hat matrix properties

The hat matrix maps responses to fitted values; it is symmetric and idempotent, its trace equals the number of parameters, and its diagonal entries are leverages.

Regression ANOVA F test

Under the null that all slope coefficients are zero, the mean square regression divided by the mean square error follows an F distribution with p−1 and n−p degrees of freedom.

Adjusted coefficient of determination

Adjusted R² equals one minus (1−R²)(n−1)/(n−p); it penalizes added regressors and can decrease when a useless variable enters the model.

Wilcoxon rank-sum test

A distribution-free test comparing two independent samples by ranking pooled observations; it is sensitive to shifts in location and uses a normal approximation for large samples.

Kolmogorov–Smirnov statistic

The supremum distance between the empirical and hypothesized cumulative distribution functions; its null distribution is distribution-free when the hypothesized law is continuous.

Bootstrap percentile interval

Resample the data with replacement many times, recompute the statistic, and take the α/2 and 1−α/2 quantiles of the resampled values as interval endpoints.

Conjugate Beta–Binomial updating

A Beta prior with parameters a and b combined with binomial data of x successes in n trials yields a Beta posterior with parameters a+x and b+n−x.

Posterior mean under squared error

The Bayes estimator minimizing expected squared error is the posterior mean; absolute error gives the posterior median and zero-one loss gives the posterior mode.

Poisson process merging

Independent Poisson processes with rates λ₁ and λ₂ superpose into a Poisson process of rate λ₁+λ₂; splitting events independently yields independent Poisson processes.

Stationary distribution condition

A probability vector π is stationary for transition matrix P when π equals πP; an irreducible aperiodic finite chain converges to this unique stationary distribution.