Relation between Binomial and Poisson Distributions

• Binomial distribution

Model for number of success in n trails where P(success in any one trail) = p.

• Poisson distribution is used to model rare occurrences that occur on average at rate λ per time interval. Can think of “rare” occurrence in terms of p Æ 0 and n Æ ∞. Take these limits so that λ = np.

• So we have that

week 5 1 Continuous Probability Spaces

• Ω is not countable. • Outcomes can be any real number or part of an interval of R, e.g. heights, weights and lifetimes. • Can not assign probabilities to each outcome and add them for events. • Define Ω as an interval that is a subset of R. • F – the event space elements are formed by taking a (countable) number of intersections, unions and complements of sub-intervals of Ω. • Example: Ω = [0,1] and F = {A = [0,1/2), B = [1/2, 1], Φ, Ω}

week 5 2 How to define P ?

• Idea - P should be weighted by the length of the intervals. - must have P(Ω) = 1 - assign 0 probability to intervals not of interest.

•For Ω the real line, define P by a (cumulative) distribution function as follows: F(x) = P((- ∞, x]).

• Distribution functions (cdf) are usually discussed in terms of random variables.

week 5 3 Recalls

week 5 4 Cdf for Continuous Probability Space

• For continuous probability space, the probability of any unique outcome is 0. Because, P({ω}) = P((ω, ω]) = F(ω) - F(ω) = 0.

•The intervals (a, b), [a, b), (a, b], [a, b] all have the same probability in continuous probability space.

• Generally speaking, – discrete random variable have cdfs that are step functions. – continuous random variables have continuous cdfs.

week 5 5 Examples

(a) X is a random variable with a uniform[0,1] distribution. The probability of any sub-interval of [0,1] is proportional to the interval’s length. The cdf of X is given by:

(b) Uniform[a, b] distribution, b > a. The cdf of X is given by:

week 5 6 Formal Definition of continuous random variable

• A random variable X is continuous if its distribution function may be written in the form

for some non-negative function f.

• fX(x)is the (Probability) Density Function of X.

• Examples are in the next few slides….

week 5 7 The Uniform distribution

(a) X has a uniform[0,1] distribution. The pdf of X is given by:

(b) Uniform[a, b] distribution, b > a. The pdf of X is given by:

week 5 8 Facts and Properties of Pdf • If X is a continuous random variable with a well-behaved cdf F then

• Properties of Probability Density Function (pdf)

Any function satisfying these two properties is a probability density function (pdf) for some random variable X.

• Note: fX (x) does not give a probability. • For continuous random variable X with density f

week 5 9 The Exponential Distribution

• A random variable X that counts the waiting time for rare phenomena has Exponential(λ) distribution. The parameter of the distribution λ = average number of occurrences per unit of time (space etc.). The pdf of X is given by:

• Questions: Is this a valid pdf? What is the cdf of X?

• Note: The textbook uses different parameterization λ = 1/θ. • Memoryless property of exponential random variable:

week 5 10 The Gamma distribution

• A random variable X is said to have a gamma distribution with parameters α > 0 and λ > 0 if and only if the density function of X is

⎧ x xe −− 1λααλ ⎪ 0 x ∞≤≤ X ()xf = ⎨ Γ()α ⎪ ⎩ 0 otherwise where

• Note: the quantity г(α) is known as the gamma function. It has the following properties: – г(1) = 1 – г(α + 1) = αг(α) – г(n) = (n – 1)! if n is an integer.

week 5 11 The Beta Distribution

• A random variable X is said to have a beta distribution with parameters α > 0 and β > 0 if and only if the density function of X is

week 5 12 The Normal Distribution

• A random variable X is said to have a normal distribution if and only if, for σ > 0 and -∞ < μ < ∞, the density function of X is

• The normal distribution is a symmetric distribution and has two parameters μ and σ. • A very famous normal distribution is the Standard Normal distribution with parameters μ = 0 and σ = 1. • Probabilities under the standard normal density curve can be done using Table III on 574 in the text book. • Example:

week 5 13 Example

• Kerosene tank holds 200 gallons; The model for X the weekly demand is given by the following density function

• Check if this is a valid pdf.

• Find the cdf of X.

week 5 14 Summary of Discrete vs. Continuous Probability Spaces

• All probability spaces have 3 ingredients: (Ω, F, P)

week 5 15