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## 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

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

## 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