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1 A Review of Probability Models Dr. Jason Merrick Bernoulli Distribution The simplest form of random variable. 0.7 – Success/Failure – Heads/Tails P ( X 1) p P ( X 0) 1 p 0.6 0.5 P(X=x) • 0.4 0.3 0.2 0.1 0 E[ X ] p 0 1 X Var( X ) p(1 p ) Review of Probability Models C5/2 Binomial Distribution The number of successes in n Bernoulli trials. 0.3 – Or the sum of n Bernoulli random variables. n x P( X x ) p (1 p)n x x 0.2 P(X=x) • 0.1 E[ X ] np 0 Var( X ) np(1 p ) 0 1 2 3 4 5 6 7 8 9 10 X Review of Probability Models C5/3 Geometric Distribution The number of Bernoulli trials required to get the first success. 0.7 0.6 0.5 P( X x) p x (1 p)n x 1 E[ X ] p Var( X ) (1 p) p2 P(X=x) • 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 X Review of Probability Models C5/4 Poisson Distribution The number of random events occurring in a fixed interval of time – Random batch sizes – Number of defects on an area of material P( X x ) x x! E[ X ] e 0.3 0.2 P(X=x) • 0.1 0 0 Var( X ) 1 2 3 4 5 6 7 8 9 10 X Review of Probability Models C5/5 Exponential Distribution Model times between events – – – – Times between arrivals Times between failures Times to repair Service Times f ( x) 1 e x / E[ X ] 0.5 0.4 0.3 f(x) • 0.2 0.1 Var( X ) 2 • 0 Memoryless 0 2 4 6 8 10 X f ( x y | X y ) f ( x) Review of Probability Models C5/6 Erlang Distribution • The sum of k exponential random variables 1 f ( x) k x k 1e x / (k 1)! 0.2 0.15 • Var( X ) k 2 f(x) E [ X ] k Gives more flexibility than exponential 0.1 0.05 0 0 2 4 6 8 10 X Review of Probability Models C5/7 Gamma Distribution A generalization of the Erlang distribution, is not required to be integer f ( x) 1 1 x / x e ( ) E [ X ] • • 1.5 Var( X ) 2 0.5 1 f(x) • 1 0.5 2 More flexible Has exponential tail 0 0 1 2 3 4 5 X Review of Probability Models C5/8 Weibull Distribution • • Commonly used in reliability analysis The rate of failures is ( x / ) 1.5 1 ( x / ) f ( x) x e 1 f(x) E[ X ] 0.5 1 0.5 2 2 2 1 1 Var( X ) 2 2 2 1 0 Review of Probability Models 0 1 2 3 4 5 X C5/9 Normal Distribution The distribution of the average of iid random variables are eventually normal f ( x) E[ X ] • 1 2 2 e 1 2 2 0.45 x 2 Var( X ) 2 Central Limit Theorem 0.3 f(x) • 0.15 0 0 2 4 6 8 10 X Review of Probability Models C5/10 Log-Normal Distribution Ln(X) is normally distributed. – Used to model quantities that are the product of a large number of random quantities – Highly skewed to the right. 1 (ln( x ) ) 2 / 2 2 f ( x) e x 2 E[ X ] e 2 0.4 0.3 f(x) • 0.2 0.1 /2 0 0 Var( X ) e 2 2 1 2 3 4 5 X 2 (e 1) Review of Probability Models C5/11 Triangular Distribution Used in situations were there is little or no data. – Just requires the minimum, maximum and most likely value. 2( x a ) f ( x) , axm ( m a )( b a ) 2( b x ) , m xb (b m)( b a ) 0, otherwise E [ X ] ( a b) / 2 0.3 0.2 f(x) • 0.1 0 0 1 2 3 4 5 6 7 8 9 X Var( X ) (b a )2 / 12 Review of Probability Models C5/12 10 Beta Distribution Again used in no data situations. 3 – Bounded on [0,1] interval. – Can scale to any interval. – Very flexible shape. 2.5 2 ( ) 1 f ( x) x (1 x ) 1 ( )( ) E[ X ] f(x) • 1.5 1 Var( X ) ( )2 ( 1) 0.5 0 0 0.25 0.5 0.75 1 X Review of Probability Models C5/13 Homogeneous Poisson Process • The number of events happening up to time t is Poisson distributed with rate t – The number of events happening in disjoint time intervals are independent – The time between events are then independent and identically distributed exponential random variables with mean 1/ – Combining two Poisson processes with rates and gives a Poisson process with rate + – Choosing events from a Poisson process with probability p gives a Poisson process with rate p – A homogeneous Poisson process is stationary Review of Probability Models C5/14 Renewal Process • If the time between events are independent and identically distributed then the number of events happening over time are a renewal process. – The homogeneous Poisson process is a renewal process with exponential inter-event times – One could also choose the inter-event times to be Weibull distributed or gamma distributed – Most arrival processes are modeled using renewal processes – Easy to use as the inter-event times are a random sample from the given distribution – A renewal process is stationary Review of Probability Models C5/15 Non-stationary Arrival Processes • External events (often arrivals) whose rate varies over time – – – – Lunchtime at fast-food restaurants Rush-hour traffic in cities Telephone call centers Seasonal demands for a manufactured product • It can be critical to model this nonstationarity for model validity – Ignoring peaks, valleys can mask important behavior – Can miss rush hours, etc. • Good model: – Non-homogeneous Poisson process Review of Probability Models C5/16 Non-stationary Arrival Processes (cont’d.) • Two issues: • – How to specify/estimate the rate function – How to generate from it properly during the simulation (will be discussed in Chapters 8, 11 …) Several ways to estimate rate function — we’ll just do the piecewise-constant method – Divide time frame of simulation into subintervals of time over which you think rate is fairly flat – Compute observed rate within each subinterval – Be very careful about time units! • Model time units = minutes • Subintervals = half hour (= 30 minutes) • 45 arrivals in the half hour; rate = 45/30 = 1.5 per minute Review of Probability Models C5/17