Random variable Theory incomplete

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Random variable Theory(incomplete)

• Summarized by
• Neetesh Purohit
• Lecturer, IIIT,
• Allahabad, UP, India
• http//profile.iiita.ac.in/np/
• 0532 2922236 (O), 0532 2922347 (R)

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Source

• Probability and statistics with reliability,
  Queuing and Computer Science Applications, IInd
  Edition, authored by K. S. Trivedi, John Wiley.
• Communication Systems, IV Edition, A. B.
  Carlson and others, McGraw Hills.

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• So far as laws of mathematics refer to reality
  they are not certain and so far as they are
  certain they do not refer to reality.
• - Albert Einstein

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Random variable X(s)

• Random variable is a rule
• X, that assigns a real number
• to each sample point, s, of a
• sample space

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Discrete Random Variable and Probability Mass
Function

• Tossing of a coin
• Transmission of Digital signals
• Access of Cache memory by the processor

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Continuous random variable
X
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CDF and PDF

• F (x) P(Xlt x)
• Draw F (x) Vs X for experiment of tossing of two
  coins.
• What is the intuitive shape of F (x) for previous
  example of disk?
• d/dx F(x) p(x)

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Properties of PDF

• Integration of p(x) over entire range 1
• P (altXltb) F(b) F(a)
• Bays theorem is applicable

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Problem on joint PDF

• If random variables B and C are independent and
  uniformly distributed over the range (0,1). Find
  the probability that the roots of following
  equation are real
• x2 2Bx C 0

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Transformation of random variables

• If X is a r. v. then Z g(x) will also be a
  random variable.
• p(z)dz p(x)dx over the range of
  monotonic relationship between X and Z.
• Find pdf of Z if it is given that ZcosX and X is
  uniformly distributed angle over (0,2pi)

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Statistical measures

• Expectation operation
• E g (x) summation/integration of
  g(xi)pi(x) over entire range of X (if X is
  discrete/cont).
• Mean
• Mean E X mx
• Derive an expression for mean for a discrete
  random variable in terms of PMF.

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Searching problem

• Names of students are stored in a table. Someone
  has a list of same students but in random order.
  He wants to search the location of a particular
  name in the table.
• If he starts from one end of the table and
  compare the given names with each stored value in
  a consecutive manner
• calculate average number of comparisons he is
  required to perform.

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• Sample mean
• sum of all observations divided by the total
  number of observations
• Always exists and is unique
• Mean gives equal weight to all observations

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Common Misuses of Means

• Usefulness of mean depends on the number of
  observations and the variance
• E.g. two response time samples 10 ms and 1000
  ms. Mean is 505 ms! Correct index but useless.
• Using mean without regard to skewness

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• Sample median
• - median is that value of X at which F(x)0.5
• list observations in an increasing order the
  observation in the middle of the list is the
  median
• Always exists and is unique

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Sample Mode

• plot histogram from the observations find
  bucket with peak frequency the middle point of
  this bucket is the mode
• Mode may not exists (e.g., all sample have equal
  weight)
• More than one mode may exist (i.e. bimodal)
• If only one mode then distribution is unimodal

mode
mode
mode
mode
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Which measure is suitable?

• Is data categorical?
• Yes use mode
• e.g. most used resource in a system
• Is total of interest?
• Yes use mean
• e.g. total response time for Web requests
• Is distribution skewed?
• Yes use median
• No use mean. Why?

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• E XY EX EY
• Var (X) E (x-mx)2 E X2 (EX)2
• Standard deviation sqrt of Var (X)
• Prove that Var XY Var X Var Y if X and
  Y are independent random variables.

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Small vs. Large Variance
Density functions for continuous random
variables with large and small variances (Source
LK00, Fig 4.6)
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• Covariance (X,Y) E (X-mx)(Y-my)
• The covariance is a measure of the dependence
  between X and Y. Note that Cov(X, X) V(X).
• Cov(X, Y) X and Y are
• 0 uncorrelated
• gt 0 positively correlated
• lt 0 negatively correlated
• Independent random variables are also
  uncorrelated but sometimes vice versa may not be
  true.

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Poisson Distribution

• Poisson distribution describes many random
  processes quite well and is mathematically quite
  simple.
• where a gt 0, pdf and cdf are
• E(X) a V(X)

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Poisson Distribution

• Example A computer repair person is beeped
  each time there is a call for service. The
  number of beeps per hour Poisson (a 2 per
  hour).
• The probability of three beeps in the next hour
• p(3) e-223/3! 0.18
• also, p(3) F(3) F(2) 0.857-0.6770.18
• The probability of two or more beeps in a 1-hour
  period
• p(2 or more) 1 p(0) p(1)
• 1 F(1)
• 0.594

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Uniform Distribution

• A random variable X is uniformly distributed on
  the interval (a,b), U(a,b), if its pdf and cdf
  are
• Properties
• P(x1 lt X lt x2) is proportional to the length of
  the interval F(x2) F(x1) (x2-x1)/(b-a)
• E(X) (ab)/2 V(X) (b-a)2/12
• U(0,1) provides the means to generate random
  numbers, from which random variates can be
  generated.

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Normal Distribution

• A random variable X is normally distributed has
  the pdf
• Mean
• Variance
• Denoted as X N(m,s2)
• Special properties
• 
  .
• f(m-x)f(mx) the pdf is symmetric about m.
• The maximum value of the pdf occurs at x m the
  mean and mode are equal.

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Normal Distribution

• Evaluating the distribution
• Use numerical methods (no closed form)
• Independent of m and s, using the standard normal
  distribution
• Z N(0,1)
• Transformation of variables let Z (X - m) / s,
  

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Normal Distribution

• Example The time required to load an oceangoing
  vessel, X, is distributed as N(12,4)
• The probability that the vessel is loaded in less
  than 10 hours
• Using the symmetry property, F(1) is the
  complement of F (-1)