1. week 51 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  0 and n  ∞. Take these limits so that λ = np. So we have that

2. week 52 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}

3. week 53 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.

4. week 54 Recalls

5. week 55 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.

6. week 56 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:

7. week 57 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. f X (x)is the (Probability) Density Function of X. Examples are in the next few slides….

8. week 58 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:

9. week 59 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: f X (x) does not give a probability. For continuous random variable X with density f

10. week 510 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:

11. week 511 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 where Note: the quantity г(α) is known as the gamma function. It has the following properties: –г(1) = 1 –г(α + 1) = α г(α) –г(n) = (n – 1)! if n is an integer.

12. week 512 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

13. week 513 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:

14. week 514 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.

15. week 515 Summary of Discrete vs. Continuous Probability Spaces All probability spaces have 3 ingredients: (Ω, F, P)