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CPSC 531 Systems Modeling and Simulation Continuous Random Variables Dr. Anirban Mahanti Department of Computer Science University of Calgary mahanti@cpsc.ucalgary.ca Definitions • A random variable X is said to be continuous if there exists a non-negative function f(x), ∀x∈(−∞,∞), with the property that for any set A of real numbers: P({ X ∈ A}) = ∫ f ( x)dx A • f(x) is called the “probability density function” (PDF) of X Continuous Random Variable 2 Properties of PDF f ( x) ≥ 0, ∀x ∞ ∫ f ( x )dx = 1 −∞ i.e., area under the curve equals one. b P ({a ≤ X ≤ b}) = ∫ f ( x)dx a i.e., B = [a, b] and we want to find P{ X ∈ B}. Continuous Random Variable 3 1 Properties of PDF (continued) Continuous distributions assign 0 value to individual values : a P ({ X = a}) = ∫ f ( x)dx = 0 a Consequence of the above property : P({a ≤ X ≤ b}) = P ({a < X < b}) = P ({a ≤ X < b}) = P ({a < X ≤ b}) Continuous Random Variable 4 Cumulative Distribution Function • The CDF FX(⋅) of a continuous random variable X with PDF fX(⋅) can be obtained as follows: FX ( x) = P({ X ∈ (−∞, x]}) x = ∫ f X (t ) dt −∞ Continuous Random Variable 5 CDF - PDF Relationship • The PDF can be obtained from the CDF and vice versa: FX' ( x) = dFX ( x) = f X ( x) dx • Distribution of a continuous random variable can be represented using either the PDF or the CDF. Continuous Random Variable 6 2 The Uniform Distribution • A random variable X is said to be uniformly distributed on the interval [a, b], a < b, if the probability of selecting a point along the interval [a, b] is equally likely at all portions of the interval. • We can say that the probability that X will belong to a particular sub-interval of [a, b] is proportional to the length of that sub-interval. Continuous Random Variable 7 PDF and CDF of Uniform R.V. • The CDF of X is: • The PDF of a uniform random variable X in the interval [a, b] is: ⎧0, x ≤ a ⎪x−a ⎪ F ( x) = ⎨ ,a< x<b ⎪b − a ⎪⎩1, x ≥ b ⎧ 1 , a< x<b ⎪ f ( x) = ⎨b − a ⎪⎩0, otherwise How did we get F(x)? x F ( a < x < b) = x dt dt x−a =∫ = b − a b − a b −a −∞ a ∫ Continuous Random Variable 8 Uniform R.V. PDF and CDF PDF of Uniform R.V. (a=1, b=3) CDF of Uniform R.V. (a=1, b=3) 1 1 f(x) 0.5 F(x) 0 0.5 0 0 1 2 3 4 0 x 1 2 3 4 x Continuous Random Variable 9 3 Exponential Distribution • A continuous random variable X is exponentially distributed with parameter β if it has the following PDF: ⎧⎪β e − β x , x ≥ 0 f X ( x) = ⎨ ⎪⎩ 0, otherwise • The CDF for the exponential distribution is: x<0 ⎧⎪ 0, FX ( x) = ⎨ −β x ⎪⎩1 − e , x≥0 Continuous Random Variable 10 Exponential Models • This distribution has been used to model: • Inter-arrival times between IP packets • Inter-arrival times between calls at a call centre • Inter-arrival times between web sessions from a web client • Service time distributions • Lifetime of products • Widely used in queuing theory Continuous Random Variable 11 Exponential PDF and CDF CDF of Exponential Distribution PDF of Exponential Distribution 1 4 β=0.5 β=1.0 3 β=2.0 β=4.0 f(x) 2 0.8 0.6 F(x) 0.4 1 0.2 0 0 β=2.0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 x x Continuous Random Variable 12 4 Memory-less Property of Exponential Distribution • Suppose inter-arrival times of IP packets are modelled using Exponential distribution. The memory-less property states that the distribution of the expected time to a packet arrival is independent of the duration there have been no packet arrivals • Suppose X is an exponentially distributed r.v. and X ≥ t (i.e., no arrivals for time t or less). Then, P({ X ≥ t+h | X ≥ t }) = P({ X ≥ h }) Continuous Random Variable 13 Memory-less Property of Exponential Distribution (cont.) • Regardless of the duration of no packet arrivals, the probability that an arrival will occur in the next h time units remains the same. P({ X ≥ t + h | X ≥ t}) = = 1 − [1 − e − β (t + h ) ] 1 − [1 − e − β t ] = P({ X ≥ h}) P({ X ≥ t + h}) P({ X ≥ t}) = e−β h • Exponential distributions are the only continuous distributions with the memory-less property. Continuous Random Variable 14 Normal Distribution • X is a normal random variable with mean μ and variance σ2 if X has the following PDF: −( x − μ ) 2 1 f X ( x) = 2π σ e 2σ 2 , −∞ < x < ∞ • The CDF of a normal distribution is: 1 FX ( x) = P ({ X ≤ x}) = 2π σ x ∫e − (t − μ ) 2 2σ 2 dt −∞ • There is no closed form for FX(x). Continuous Random Variable 15 5 PDF of Normal Distribution PDF of Normal Distribution (μ = 0) 1 0.8 σ=1 σ=0.5 0.6 σ=2 f(x) 0.4 0.2 0 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 x This PDF has a “bell” shape with “peak” at x=0. Continuous Random Variable 16 Why is Normal Distribution Important? • Appropriate or useful for modelling many realworld phenomena • IQs of randomly selected people • Height of people selected from a homogeneous population • Mathematical convenience: many statistical analysis methods assume a normal distribution • Central Limit Theorem: The sample mean of a large, random sample from any distribution with finite variance will be approximately normally distributed. Continuous Random Variable 17 More on Normal Distribution • Terminology: also called the “Gaussian” distribution, after German mathematician and scientist Carl Friedrich Gauss (1777-1855) • In a normal distribution • 68% of the values are within 1 standard deviation of the mean • Approximately 95% of the values are within 2 standard deviations of the mean • Approximately 99% of the values are within 3 standard deviations of the mean Continuous Random Variable 18 6 Even more on Normal Distribution • If X is normally distributed with parameters μ and σ2, then Z= (X − μ) σ is normally distributed with parameters 0 and 1. Z is called the “standard normal distribution”. Continuous Random Variable 19 Computing CDF of Normal Distribution • Normal distribution has no closed form for F(x). How to compute P({a < X < b})? • Transform X to standard normal distribution Z and use tables • If X ~ N ( μ , σ 2 ) then Z = ( X − μ) σ is N (0,1) Continuous Random Variable 20 Omnipresence of Normal Distribution! • Normal distribution is everywhere, even on a bank note. Continuous Random Variable 21 7