Random Variables

A random variable is an object, usually a number of a vector, whose value depends on the outcome of a random experiment Definition Let –> S be the sample space of a probability model –> E be a Set A function: X : S –> E s –> X(s) is called a random variable with values in E

To every B is a subset of E we associate P(XϵB) = µx(B) It can be shown that µx is a probability on the subsets of E, called the distribution of X

A random variable X is said to be discrete if it can attain finitely or countably many values. More precisely, there exists a N is a subset of E finite or countable. Such that P(XϵN) = 1

a set N is countable if its elements can be enumerated N = {x1, x2, …, xn} –> countable set: - integer numbers - rational numbers –> not countable set: - real numbers

If X is a discrete random variable, define px: E –> [0,1] x –> P(X=x) px(x) = P(X=x) px is called probability mass function or discrete density By using ͳ-additivity of the probability we have for A is a subset of E µx(A) = P(xϵA) = Σ P(X=x) µx(A) = Σ px(x)

Let X be a discrete random variables with values in R Denote by N = {x1, x2, …, xn} the finite or countable sets of its values The expected value of X is defined by E(X)= Σxi px(xi)= Σx px(x)

1) Constant random variables 2) Invariance by linear transformation 3) E[f(x)]=Σ f(xi) px(xi) 4) Linearity

Let C be a real number. A constant can be viewed as a random variable that attains the value c with probability 1. So: E(c) = c * 1 = c

Suppose –> X is a: - discrete - real valued - random variable –> a,b ϵ R Then E(aX + b) = a * E(x) + b

Suppose X is a: - discrete random variable - values in the set E Let f: E –> R Remark: a random variable is a function that goes from a sample space to E x: S –> E S -x-> E -f-> R 2 function related one after another one E[f(x)] = Σ f(xi) px(xi) How to read this formula If I have a discrete random variable with his density and I consider a generic transformation of this random variable than I can know the expectation value of this transformation by knowing the density of the discrete random variable

Suppose: X and Y are two real valued variables defined on the same sample space S If a,b ϵ R E(aX+bY) = aE(X) + bE(Y)

Suppose X is a discrete real valued random variable For m>1 consider the random variable X^m By property 3 E(X^m)= Σ xi^m px(xi)

High variance indicates that X attains values far from its expectation with a significance probability Var(x) = Σ (xi - µ)² px(xi)

SD(x) = √Var(x)

1) Var(x) = E(x²) - (E(x))² 2) Var(aX+b) = a² * Var(X)

Only take values 0 and 1 Density –> X~Be(p) P(X=1) = p P(X=0) = 1-p E(X) = p Var(X) = p(1-p)

Consider the random experiment of independent trials with probability of success p Denote by X the number of successes in the first n trials px(k) = P(X=k) =(n) p^k (1-p)^(n-k) (k) Density –> X~Bin(n,p) E(X) = np Var(X) = np(1-p) E(X^m)=npE[(y+1)^(m-1)] (With Y~Bin(n-1,p))

We say that X is a Poisson random variable with parameter λ>0, and we write X~Pois(λ) px(n) = P(X=n) = (e^-λ) * (λ^n)/(n!) Note: this is a density since Σ (λ^n)/(n!) = (e^λ) They emerge as approximation of Binomial random variables This bond is used when: ▪ p«1 and n»1 ▪ np = λ is neither small nor larger ▪ np²«1 E(X) = λ Var(X) = λ E(X^m) = λ E((X+1)^(m-1)) If m = 2 then E(X²) = λ²+λ

Consider repeated independent trials with probability of success p, and denote by X the trial corresponding to the first success X = n <–> the first n-1 trials are losses and the nth trial is a success px(n) = P(X=n) = p * (1-p)^(n-1) X~Geo(p) E(X) = 1/p Var(X) = (1-p)/p²

P ( X>n+k|X>n ) = P ( X>k )