CPSC 531: Probability Review 1
CPSC 531:Probability & Statistics: Review II
Instructor: Anirban MahantiOffice: ICT 745Email: [email protected] Location: TRB 101Lectures: TR 15:30 – 16:45 hoursClass web page:
http://pages.cpsc.ucalgary.ca/~mahanti/teaching/F05/CPSC531
Notes derived from “Probability and Statistics” by M. DeGroot and M. Schervish, Third edition, Addison Wesley, 2002, and
“Discrete-event System Simulation” by Banks, Carson, Nelson, and Nicol, Prentice Hall, 2005.
CPSC 531: Probability Review 2
Objective and Outline The world the model-builder sees is probabilistic
rather than deterministic. Some statistical model might well describe the
variations.
An appropriate model can be developed by sampling the phenomenon of interest: Select a known distribution through educated guesses Make estimate of the parameters Test for goodness of fit
Goal is to review: Random variables Discrete and continuous random variables Cumulative distribution functions Expectation, variance, etc.
CPSC 531: Probability Review 3
Random Variables A random variable is a real-valued mapping
defined on a sample space.
Suppose that X is a random variable defined on space S, then X assigns a real-number X(s) to each possible outcome s є S.
Typically, X, Y, Z etc denote random variables; x, y, z, etc denote values attained by random variables.
Example: Rolling a pair of dice. Let X be the random variable corresponding to the sum of the dice on a roll. If we think of the sample points as a pair (i, j), where i = value rolled by the first dice and j = value rolled by the second dice, we have:
X(s) = i+j
CPSC 531: Probability Review 4
Discrete Random Variables A random variable X is said to be discrete if the
number of possible values of X is finite, or at most, an infinite sequence of different values.
Example: Consider jobs arriving at a job shop.• Let X be the number of jobs arriving each week at a job
shop.• S = possible values of X (range space of X) = {0,1,2,…} • p(xi) = probability the random variable is xi = P(X = xi)
p(xi), i = 1,2, … must satisfy:
The collection of pairs [xi, p(xi)], i = 1,2,…, is called the probability distribution of X, and p(xi) is called the probability mass function (pmf) of X.
The pmf is referred to as “probability function” in some texts
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i allfor ,0)( 1.
i i
i
xp
xp
CPSC 531: Probability Review 5
Discrete Random Variables Consider a random variable X that takes on values
1, 2, 3, and 4 with probabilities 1/6, 1/3, 1/3, and 1/6, resp.
1 2
p(x)
3 4
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00 x
CPSC 531: Probability Review 6
Continuous Random Variables X is a continuous random variable if there exists a non-
negative function f(x) such that for any set of real numbers A є S
The probability that X lies in the interval [a,b] is given by:
f(x), denoted as the pdf of X, satisfies:
Properties
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00
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CPSC 531: Probability Review 7
Continuous Random Variables
Example: Life of an inspection device is given by X, a continuous random variable with pdf:
X has an exponential distribution with mean 2 years Probability that the device’s life is between 2 and 3 years
is:
otherwise ,0
0 x,2
1)(
2/xexf
14.02
1)32(
3
2
2/ dxexP x
CPSC 531: Probability Review 8
Cumulative Distribution Function The cumulative distribution function (cdf) of a random
variable X is a function F(x), defined for each real number x: F(x) = P(X <= x) for -∞ < x < ∞
If X is discrete, then
If X is continuous, then Properties
All probability question about X can be answered in terms of the cdf, e.g.:
xx
i
i
xpxF all
)()(
xdttfxF )()(
0)(lim 3.
1)(lim 2.
)()( then , If function. ingnondecreas is 1.
xF
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bFaFbaF
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x
baaFbFbXaP allfor ,)()()(
CPSC 531: Probability Review 9
Cumulative Distribution Function Example: An inspection device has cdf:
The probability that the device lasts for less than 2 years:
The probability that it lasts between 2 and 3 years:
2/
0
2/ 12
1)( xx t edtexF
632.01)2()0()2()20( 1 eFFFXP
145.0)1()1()2()3()32( 1)2/3( eeFFXP
CPSC 531: Probability Review 10
Expectation The expected value of X is denoted by E(X)
If X is discrete
If X is continuous
The mean, μ, is the 1st moment of X A measure of the central tendency
Properties: E(cX) = cE(X), where c is a constant E(Y) = aE(X) + b, where Y=aX+b, a & b are constants E(X + Y) = E(X) + E(Y) regardless of whether X and Y are
independent E(X.Y) = E(X).E(Y) if X & Y are independent
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)()(
CPSC 531: Probability Review 11
Variance The variance of X is denoted by V(X) or
var(X) or 2
Definition: V(X) = E[(X – E[X]2] Also, V(X) = E(X2) – [E(x)]2
The variance is a measure of the dispersion or spread of a random variable about its mean
The standard deviation of X is denoted by Definition: square root of V(X) Expressed in the same units as the mean
Properties: V(cX) = c2V(X) V(X + Y) = V(X) + V(Y) if X, Y are independent
CPSC 531: Probability Review 12
Density functions for continuous random variables with large and small variances (Source LK00, Fig 4.6)
µ µ
σ2
largeσ2
small
X X X X
Small vs. Large Variance
CPSC 531: Probability Review 13
Expectations and Variance (example) Example: The mean of life of the previous inspection
device is:
To compute variance of X, we first compute E(X2):
Hence, the variance and standard deviation of the device’s life are:
22/2
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2/
00
2/
dxexdxxeXE xx xe
82/22
1)(
0
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2/22
dxexdxexXE xx ex
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XV
XV
CPSC 531: Probability Review 14
Joint Distributions Let X and Y each have a discrete
distribution. Then X and Y have a discrete joint distribution if there exists a function p(x,y) such that:
p(x,y) = P[X=x and Y=y] Random variables X and Y are jointly
continuous if there exists a non-negative function f(x,y) called the joint probability density function of X and Y, such that for all sets of real numbers A and B
P(X є A, Y є B) = ∫ ∫f(x,y)dxdyB A
CPSC 531: Probability Review 15
Covariance
The covariance between the random
variables X and Y, denoted by Cov(X, Y), is
defined by Cov(X, Y) = E{[X - E(X)][Y -
E(Y)]}
= E(XY) - E(X)E(Y)
The covariance is a measure of the
dependence between X and Y. Note that
Cov(X, X) = V(X).
CPSC 531: Probability Review 16
Covariance
Cov(X, Y) X and Y are
= 0 uncorrelated> 0 positively correlated< 0 negatively correlated
Independent random variables are also uncorrelated.
CPSC 531: Probability Review 17
Statistical Models Application areas where statistical models
find widespread use: Queueing systems Inventory and supply-chain systems Reliability and maintainability Limited data
CPSC 531: Probability Review 18
Queueing Systems In a queueing system, interarrival and service-time
patterns can be probabilistic (e.g., our M/M/1 example).
Sample statistical models for interarrival or service time distribution: Exponential distribution: if service times are completely
random Normal distribution: fairly constant but with some random
variability (either positive or negative) Truncated normal distribution: similar to normal
distribution but with restricted value. Gamma and Weibull distribution: more general than
exponential (involving location of the modes of pdf’s and the shapes of tails.)
CPSC 531: Probability Review 19
Inventory and supply chain In realistic inventory and supply-chain systems,
there are at least three random variables: The number of units demanded per order or per time
period The time between demands The lead time
Sample statistical models for lead time distribution: Gamma
Sample statistical models for demand distribution: Poisson: simple and extensively tabulated. Negative binomial distribution: longer tail than Poisson
(more large demands). Geometric: special case of negative binomial given at
least one demand has occurred.
CPSC 531: Probability Review 20
Reliability and maintainability Time to failure (TTF)
Exponential: failures are random Gamma: for standby redundancy where each
component has an exponential TTF Weibull: failure is due to the most serious of a
large number of defects in a system of components
Normal: failures are due to wear
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