AkashNotes
Maximum Likelihood Estimation
Let X1, . . . Xn be IID with PDF f(x; θ). The likelihood function is defined byThe log-likelihood function
Parametric Inference and Method of moments
Till now, we have covered the estimation of statistical functionals i.e. functions of CDF – Fx. They are Non-parametric Inference.
Bootstrap Confidence Interval
There are three main methods of constructing Bootstrap confidence interval. Normal Interval Pivotal Interval Percentile Interval Normal Interval The normal
Normal (Gaussian) Distribution: Everything you need to know
Normal or Gaussian distribution is one of the most widely used distribution functions in statistics. This post covers all the
Bootstrap
Bootstrap is a non-parametric method for estimating accuracy defined in terms of standard error, bias, variance, confidence interval, etc. Suppose
Empirical Distribution Function and Estimation of Statistical Functionals
When starting with the inference problem, the most basic is the non-parametric estimation of CDF and functions of CDF. Let
Confidence set
For a parameter θ, a 1-α confidence interval is Cn = (a, b)where a = A(X1,. . , Xn) and
Introduction to Point Estimation
Point estimation refers to the use of sample data to provide a single best guess (known as point estimate) of
Parametric and Non-Parametric models
A statistical model is a set of distribution or a set of densities. A parametric model is a statistical model
Law of Large Number and CLT
The Weak Law of Large Numbers (WLLN) If X1, X2, . . . , Xn are IID, then This theorem
Convergence of Random Variable
Pointwise or sure convergence A sequence of random variables {Xn}n∈N is said to converge point-wise or surely to X if Xn(ω) → X(ω), ∀
Inequalities
Markov’s Inequalities Let X be a non-negative random variable and suppose that E(X) exists. For any t > 0, Chebyshev’s
Variance and Covariance
Variance Variance means spread of a distribution Let X be a random variable with mean μ. The variance of X
Expectation
The expected value, or mean, or first moment, of X is defined to beassuming that the sum (or integral) is
Transformation of Random Variables
Suppose X is a random variable with PDF fX and CDF FX. Let Y = r(X) be a function of
Independent and Identical Distributed Samples
Let X = (X1, . . . , Xn) where X1, . . . , Xn are random variables. We
Independent Random Variables and Conditional Distribution
Independent Random Variables Two random variables are X and Y are independent if, for every A and B,P(X ∈ A,
Bivariate and Marginal Distribution
Joint Mass Function Remember the probability mass function definition. That is the study of one random variable. Given two discrete
Continuous Random Variables
Uniform Probability Distribution X has Uniform(a, b) distribution, written X~Uniform(a, b), ifwhere a < b. The distribution function is The
Discrete Random Variables
In this section, we are going to cover some important Discrete Random Variables. Note that we will be writing X
Introduction to Random Variables
Random Variable A random variable is a mappingX : Ω→Rthat assigns a real number X(ω) to each outcome ω Getting
Bayes’ Theorem
Partition A partition of Ω is a sequence of disjoint sets A1, A2, … such that The Law of Total
Introduction to Probability – 2
Uniform Probability distribution If Ω is finite and each outcome is equally likely then, where |A| denotes the number of elements in
Introduction to Probability
Probability quantifies uncertainty. It is a measure of “how likely” an “event” can occur. Probability is measured on a scale