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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 4 1 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 4 2 Recalls week 4 3 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 4 4 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 4 5 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 4 6 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 4 7 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 4 8 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/β. • Memoryless property of exponential random variable: week 4 9 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 e x x 1 f X x 0 0 x otherwise 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. week 4 10 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 4 11 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 4 in Appendix 3 of the text book. • Example: week 4 12 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 4 13 Summary of Discrete vs. Continuous Probability Spaces • All probability spaces have 3 ingredients: (Ω, F, P) week 4 14 Poisson Processes • Model for times of occurrences (“arrivals”) of rare phenomena where λ – average number of arrivals per time period. X – number of arrivals in a time period. • In t time periods, average number of arrivals is λt. • How long do I have to wait until the first arrival? Let Y = waiting time for the first arrival (a continuous r.v.) then we have Therefore, which is the exponential cdf. • The waiting time for the first occurrence of an event when the number of events follows a Poisson distribution is exponentialy distributed. week 4 15 Expectation • In the long run, rolling a die repeatedly what average result do you expact? • In 6,000,000 rolls expect about 1,000,000 1’s, 1,000,000 2’s etc. Average is • For a random variable X, the Expectation (or expected value or mean) of X is the expected average value of X in the long run. • Symbols: μ, μX, E(X) and EX. week 4 16 Expectation of discrete random variable • For a discrete random variable X with pmf whenever the sum converge absolutely week 4 . 17 Examples 1) Roll a die. Let X = outcome on 1 roll. Then E(X) = 3.5. 2) Bernoulli trials and . Then 3) X ~ Binomial(n, p). Then 4) X ~ Geometric(p). Then 5) X ~ Poisson(λ). Then week 4 18 Expectation of continuous random variable • For a continuous random variable X with density whenever this integral converge absolutely. week 4 19 Examples 1) X ~ Uniform(a, b). Then 2) X ~ Exponential(λ). Then 3) X is a random variable with density (i) Check if this is a valid density. (ii) Find E(X) week 4 20 4) X ~ Gamma(α, λ). Then 5) X ~ Beta(α, β). Then week 4 21