AkashNotes

Law of Large Number and CLT

The Weak Law of Large Numbers (WLLN) If X1, X2, . . . , Xn are IID, then This theorem

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), ∀

Markov’s Inequalities Let X be a non-negative random variable and suppose that E(X) exists. For any t > 0, Chebyshev’s

Variance Variance means spread of a distribution Let X be a random variable with mean μ. The variance of X

The expected value, or mean, or first moment, of X is defined to beassuming that the sum (or integral) is

Suppose X is a random variable with PDF fX and CDF FX. Let Y = r(X) be a function of

Let X = (X1, . . . , Xn) where X1, . . . , Xn are random variables. We

Independent Random Variables Two random variables are X and Y are independent if, for every A and B,P(X ∈ A,

Joint Mass Function Remember the probability mass function definition. That is the study of one random variable. Given two discrete

Uniform Probability Distribution X has Uniform(a, b) distribution, written X~Uniform(a, b), ifwhere a < b. The distribution function is The

In this section, we are going to cover some important Discrete Random Variables. Note that we will be writing X

Random Variable A random variable is a mappingX : Ω→Rthat assigns a real number X(ω) to each outcome ω Getting

A partition of Ω is a sequence of disjoint sets A1, A2, … such that The Law of Total Probability

Uniform Probability distribution If Ω is finite and each outcome is equally likely then, where |A| denotes the number of elements in

Probability quantifies uncertainty. It is a measure of “how likely” an “event” can occur. Probability is measured on a scale