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TDC 369 / TDC 432. April 2, 2003 Greg Brewster. Topics. Math Review Probability Distributions Random Variables Expected Values. Math Review. Simple integrals and differentials Sums Permutations Combinations Probability. Math Review: Sums. Math Review: Permutations. - PowerPoint PPT Presentation

### Transcript of TDC 369 / TDC 432

• TDC 369 / TDC 432April 2, 2003

Greg Brewster

• Topics Math ReviewProbability DistributionsRandom VariablesExpected Values

• Math Review Simple integrals and differentialsSumsPermutationsCombinationsProbability

• Math Review: Sums

• Math Review:Permutations Given N objects, there are N! = N(N-1)1 different ways to arrange themExample: Given 3 balls, colored Red, White and Blue, there are 3! = 6 ways to order themRWB, RBW, BWR, BRW, WBR, WRB

• Math Review:Combinations The number of ways to select K unique objects from a set of N objects without replacement is C(N,K) = Example: Given 3 balls, RBW, there are C(3,2) = 3 ways to uniquely choose 2 ballsRB, RW, BW

• Probability Probability theory is concerned with the likelihood of observable outcomes (events) of some experiment.Let be the set of all outcomes and let E be some event in , then the probability of E occurring = Pr[E] is the fraction of times E will occur if the experiment is repeated infinitely often.

• Probability Example:Experiment = tossing a 6-sided dieObservable outcomes = {1, 2, 3, 4, 5, 6}For fair die, Pr{die = 1} = Pr{die = 2} = Pr{die = 3} = Pr{die = 4} = Pr{die = 5} = Pr{die = 6} =

• Probability Pie

• Valid Probability MeasureA probability measure, Pr, on an event space {Ei} must satisfy the following:For all Ei , 0
• Probability Mass FunctionPr(Die = x)

• Mass Function = Histogram If you are starting with some repeatable events, then the Probability Mass function is like a histogram of outcomes for those events.The difference is a histogram indicates how many times an event happened (out of some total number of attempts), while a mass function shows the fraction of time an event happens (number of times / total attempts).

• Dice Roll Histogram1200 attemptsNumber of times Die = x

• Probability Distribution Function(Cumulative Distribution Function)Pr(Die
• Combining EventsProbability of event not happening: Probability of both E and F happening:IF events E and F are independent Probability of either E or F happening:

• Conditional Probabilities The conditional probability that E occurs, given that F occurs, written Pr[E | F], is defined as

• Conditional Probabilities Example: The conditional probability that the value of a die is 6, given that the value is greater than 3, is Pr[die=6 | die>3] =

• Probability Pie

• Conditional Probability Pie

• Independence Two events E and F are independent if the probability of E conditioned on F is equal to the unconditional probability of E. That is, Pr[E | F] = Pr[E].In other words, the occurrence of F has no effect on the occurrence of E.

• Random Variables A random variable, R, represents the outcome of some random event. Example: R = the roll of a die.The probability distribution of a random variable, Pr[R], is a probability measure mapping each possible value of R into its associated probability.

• Sum of Two Dice Example:R = the sum of the values of 2 diceProbability Distribution: due to independence:

• Sum of Two Dice

• Probability Mass Function:R = Sum of 2 dicePr(R = x)

• Continuous Random Variables So far, we have only considered discrete random variables, which can take on a countable number of distinct values.Continuous random variables and take on any real value over some (possibly infinite) range.Example: R = Inter-packet-arrival times at a router.

• Continuous Density Functions There is no probability mass function for a continuous random variable, since, typically, Pr[R = x] = 0 for any fixed value of x because there are infinitely many possible values for R.Instead, we can generate density functions by starting with histograms split into small intervals and smoothing them (letting interval size go to zero).

• Example: Bus Waiting TimeExample: I arrive at a bus stop at a random time. I know that buses arrive exactly once every 10 minutes. How long do I have to wait?Answer: My waiting time is uniformly distributed between 0 and 10 minutes. That is, I am equally likely to wait for any time between 0 and 10 minutes

• Bus Wait Histogram2000 attempts (histogram interval = 2 min)Waiting Times (using 2-minute buckets)

• Bus Wait Histogram2000 attempts (histogram interval = 1 min)Waiting Times (using 1-minute buckets)

• Bus Waiting TimeUniform Density Function

• Value for Density FunctionThe histograms show the shape that the density function should have, but what are the values for the density function?Answer: Density function must be set so that the function integrates to 1.

• Continuous Density Functions To determine the probability that the random value lies in any interval (a, b), we integrate the function on that interval.

So, the probability that you wait between 3 and 5 minutes for the bus is 20%:

• Cumulative Distribution FunctionFor every probability density function, fR(x), there is a corresponding cumulative distribution function, FR(x), which gives the probability that the random value is less than or equal to a fixed value, x.

• Example: Bus Waiting TimeFor the bus waiting time described earlier, the cumulative distribution function is

• Bus Waiting TimeCumulative Distribution FunctionPr(R
• Cumulative Distribution Functions The probability that the random value lies in any interval (a, b) can also easily be calculated using the cumulative distribution function

So, the probability that you wait between 3 and 5 minutes for the bus is 20%:

• Expectation The expected value of a random variable, E[R], is the mean value of that random variable. This may also be called the average value of the random variable.

• Calculating E[R] Discrete R.V.Continuous R.V.

• E[R] examples Expected sum of 2 diceExpected bus waiting time

• MomentsThe nth moment of R is defined to be the expected value of RnDiscrete:Continuous:

• Standard DeviationThe standard deviation of R, (R), can be defined using the 2nd moment of R:

• Coefficient of VariationThe coefficient of variation, CV(R), is a common measure of the variability of R which is independent of the mean value of R:

• Coefficient of VariationThe coefficient of variation for the exponential random variable is always equal to 1.Random variables with CV greater than 1 are sometimes called hyperexponential variables.Random variables with CV less than 1 are sometimes called hypoexponential variables.

• Common Discrete R.V.sBernouli random variableA Bernouli random variable w/ parameter p reflects a 2-valued experiment with results of success (R=1) w/ probability p

• Common Discrete R.V.sGeometric random variableA Geometric random variable reflects the number of Bernouli trials required up to and including the first success

• Geometric Mass Function# Die Rolls until a 6 is rolledPr(R = x)

• Geometric Cumulative Function# Die Rolls until a 6 is rolledPr(R
• Common Discrete R.V.sBinomial random variableA Binomial random variable w/ parameters (n,p) is the number of successes found in a sequence of n Bernoulli trials w/ parameter p

• Binomial Mass Function# 6s rolled in 12 die rollsPr(R = x)

• Common Discrete R.V.sUniform random variableA Uniform random variable w/ parameter set {x1 xN} is one which picks each xi value with equal probability

• Common Discrete R.V.sPoisson random variableA Poisson random variable w/ parameter models the number of arrivals during 1 time unit for a random system whose mean arrival rate is arrivals per time unit

• Poisson Mass FunctionNumber of Arrivals per second given an average of 4 arrivals per second ( = 4)Pr(R = x)

• Continuous R.V.sContinuous Uniform random variableA Continuous Uniform random variable is one whose density function is constant over some interval (a,b):

• Exponential random variableA (Negative) Exponential random variable with parameter represents the inter-arrival time between arrivals to a Poisson system:

• Exponential random variableMean (expected value) and coefficient of variation for Exponential random variable:

• Exponential DelayPoisson 4 arrivals/unit (E[R] = 0.25)Pr(R