More Con nuous Distribu ons
Keegan Korthauer Department of Sta s cs UW Madison
1 Some Common Distribu ons
Probability Distribu ons
Discrete Con nuous
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 simula ons
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
• Con nuous 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 exponen al 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
• Con nuous • 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 me 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
h p://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 emi