More Con(nuous Distribu(ons Keegan Korthauer Department of Stas(cs UW Madison 1 Some Common Distribu(ons Probability Distribu(ons Discrete Connuous Finite Infinite Finite Infinite Bernoulli and Uniform Normal Exponen(al Poisson Geometric Binomial 2 CONTINUOUS DISTRIBUTIONS Normal distribu(on Con(nuous Uniform Distribu(on Exponen(al distribu(on 3 Uniform Distribu(on • A normally distributed RV tends to be close to its mean • Instead consider a random variable X that is equally likely to take on any value in a con(nuous interval (a,b) – Example: wai(ng (me for a train if we know it will arrive within a certain (me range but have no prior knowledge beyond that • Then X is distributed uniformly on the interval (a,b) • The Uniform distribu(on is especially useful in computer simulaons 4 X ~ Uniform(a,b) • Probability Density Func(on " 1 $ if a < x < b f (x) = #b − a $ %0 otherwise • Mean and Variance a + b µ = X 2 2 2 (b − a) σ = X 12 5 Uniform PDF • Connuous f(x) • Finite • Symmetric • Constant x 6 Example – Wai(ng Time • Suppose we know that the (me a train will arrive is distributed uniformly between 5:00pm and 5:15pm • Let X represent the number of minutes we have to wait if we arrive at 5:00pm. • What is the expected (me we will wait? 7.5 minutes • What is the probability that we will wait longer than 10 minutes? 1/3 7 Infinite Wai(ng Times? • Say we want to model the life(me X of a component and we do not have an upper bound • We want to allow for the possibility that the component will last a really long (me (let X approach infinity) • Our PDF s(ll has to integrate to one, so it will have to asympto(cally decrease to zero as X goes to infinity • In this case, we can model X as an exponenal random variable with shape parameter λ • Can think of the exponen(al as the con(nuous analog to the geometric distribu(on 8 X ~ Exponen(al(λ) • Probability Density Func(on −λx $"λe if x > 0 f (x) = # %$0 otherwise • Mean and Variance 1 µ = X λ 1 σ 2 = X λ 2 9 Exponen(al Distribu(on • Connuous • Infinite • Support: x>0 • Shape determined by λ 10 Connec(on to the Poisson • Recall that the Poisson distribu(on models the number of events occurring within one unit of (me or space given some rate parameter λ • If instead of coun(ng the number of events that occur we are interested in modeling the 0me between two events, we can use the exponen(al distribu(on • Formally, if events follow a Poisson process with rate parameter λ, then the wai(ng (me T from any star(ng point un(l the next event (me is distributed Exp(λ) 11 Calling Center Example hp://www.statlect.com/uddpoi1.htm 12 Example 4.58 • A radioac(ve mass emits par(cles according to a Poisson process at a rate of 15 par(cles per minute. At some point, we start a stopwatch. • What is the probability that more than 5 seconds will elapse before the next emission? P(T>5) = 0.2865 • What is the mean wai(ng (me un(l the next par(cle is emibed? E(T) = 4 seconds 13 Lack of Memory Property • The exponen(al distribu(on does not ‘remember’ how long we’ve been wai(ng • The (me un(l the next event in a Poisson process from any star0ng point is exponen(al • The distribu(on of the wai(ng (me is the same whether the (me period just started or we have already waited t (me units • We call this memorylessness or the lack of memory property If T ~ Exp(λ) then P(T > t | T > s) = P(T > t) for s, t > 0 14 Example 4.59-60 – Circuit Life(me • The life(me of a par(cular circuit has an exponen(al distribu(on with mean 2 years. • Find the probability that the circuit lasts longer than 3 years. • Assume the circuit is already 4 years old. What is the probability that it will last more than 3 addi(onal years? 15 When λ is Unknown How can we es(mate it? • If X ~ Exp(λ) then μX = 1/λ so λ = 1/μX 1 • Then we can es(mate λ with λˆ = X • Problem - although μX is unbiased for the sample mean, 1/ μX is not a linear func(on of μX so our es(mator is biased: E(λˆ) ≈ λ + λ / n • For large samples, the bias will be negligible but for small samples we can correct for the bias with the es(mator 1 n λˆ = = corr X + X / n (n +1)X 16 When λ is Unknown What is the uncertainty in our es(mate? • Using the propagaon of error method from Chapter 3, we can es(mate it with d 1 1 1 plug in λ =λˆ 1 σ ˆ ≈ σ = = λ dX X X X 2 λ n X n • Because of the bias in the es(mate of λ, we will underes(mate the uncertainty when we have a small sample size 17 Next • Skim sec(on 4.6 (the lognormal distribu(on) and the rest of sec(on 4.8 (the gamma and Weibull distribu(ons) • Exam 1 (next Friday 2/28) covers material up to and including this lecture – Prac(ce exam handed out this Friday – Review session on Wednesday • In the next few lectures we will – finish up Chapter 4 -Probability Plots and the Central Limit Theorem – introduce the stas(cal language R 18 .