Important Random Variables Binomial: S X = { 0,1,2,..., n} Geometric: S X = { 0,1,2,... } Poisson: S X = { 0,1,2,... }

Poisson Distribution: Used for modeling number of events in an interval or set if they occur randomly and independently. If N = Number of events in an interval T, Where α = Event rate  T = Average number of events per size T Interval. e.g. Failure rate for chips in a system = 2 / year For a large number of Bernoulli trials with small success rate, p, Number of successes = Poisson

P( k hits in n attempts) = with α = np as n → ∞ and np is constant. e.g.

Examples of Poisson Distribution Applications 1) 2)

Continuous Random Variables Uniform: Gaussian: Exponential:

Mean: also called the expected value of X. If X is discrete E(X) does not exist for all random variables. It requires that:

Variance: Standard Deviation = √Var Variance measures the dispersion of X about the mean. Moments : nth Moment (X) = nth Central Moment = nth Absolute Moment = nth Generalized Moment about a =

Markov Inequality Proof:

Chebychev Inequality: Beinayme Inequality: The Chebychev inequality is a special case of this with b=m, n=2

Memoryless Random Variables: A random variable, X, is called memoryless if, for h>0 i.e. the incremental probability of x+h is independent of x. Context is meaningless. Geometric is the only memoryless discrete random variable. Exponential is the only memoryless continuous random variable.

Gaussian Random Variable The Gaussian distribution is also called the normal distribution and is often popularly referred to as the bell curve. It is found to be a good model for random variables in many real-world systems, and has many useful properties (as we will see later in the class). m f X (x) There is no closed form for F X (x) x  

Standard Gaussian Random Variable Note: The textbook uses  instead of G, but we will later use  for something else.

De Moivre-Laplace Theorem: if np(1-p) >> 1 where m = np and  = np(1-p)

Functions of Random Variables: If X is a random variable Y = g(X) is also a random variable. Any event {g(X) ≤ a} can be seen as a union of events in S X. This is called the equivalent event. e.g. g(x)g(x) a i j k x

In general, if y=g(X) has n solutions, {x k } y+dy x 1 + dx 1 y x1x1 x2x2 x 2 + dx 2 x 3 + dx 3 x3x3 Example:

Linear Case: Since Y is linear in X, there is only one solution to y=ax + b : and