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Continuous random variables

Continuous random variables. Continuous random variable Let X be such a random variable Takes on values in the real space (-infinity; +infinity) (lower bound; upper bound) Instead of using P(X= i ) Use the probability density function f X (t) Or f X (t) dt.

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Continuous random variables

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  1. Continuous random variables • Continuous random variable • Let X be such a random variable • Takes on values in the real space • (-infinity; +infinity) • (lower bound; upper bound) • Instead of using P(X=i) • Use the probability density function • fX (t) • Or fX (t)dt

  2. Cumulative function of continuous r.v. • The relationship between the • Cumulative distribution of continuous r.v. and fX • => • Properties for CDF

  3. Distribution function: properties • Properties for pdf

  4. Uniform random variable • X is a uniform random variable • Mean: • Variance:

  5. Exponential distribution • Exponential distribution • is the foundation of most of the stochastic processes • Makes the Markov processes ticks • is used to describe the duration of sthg • CPU service • Telephone call duration • Or anything you want to model as a service time

  6. Exponential random variable • A continuous r.v. X • Whose density function is given for • is said to be an exponential r.v. with parameter λ • Mean: and variance:

  7. Exponentially distributed with 1/ λ T Link between Poisson and Exponential • If the arrival process is Poisson • # arrivals per time unit follows the Poisson distribution • With parameter λ • => inter-arrival time is exponentially distributed • With mean = 1/ λ = average inter-arrival time 0 Time

  8. Proof • Number of arrivals in a t-second interval • Follows the Poisson distribution with parameter • Let denote random time of first arrival • => T is exponentially distributed

  9. Memoryless Property Proof:

  10. Example • Suppose that amount of time you spend in bank • is exponential with mean 10 min • What is the probability you spend more than 5 min in bank? • What is the probability you spend more than 15 min • Given that you are still in bank after 10 min?

  11. Hyper-exponential distributions • H2 Hn • Advantage • Allows a more sophisticated representation • Of a service time • While preserving the exponential distribution • And have a good chance of analyzing the problem λ1 p1 λ1 p λ2 p2 . . 1-p λ2 λn pn

  12. Further properties of the exponential distribution • If are independent exponential r.v. • With mean , then the pdf of is: • Gamma distribution with parameters n and • If and are independent exponential r.v. • With mean and =>

  13. Further properties of the exponential distribution (ct’d) • X1 , X2 , …, Xn independent r.v. • Xi follows an exponential distribution with • Parameter λi => fXi (t) = λieλit • Define • X = min{X1, X2, …, Xn} is also exponentially distributed • Proof • fX(t) = ?

  14. Joint distribution functions • Discrete case • One variable (pmf) • P(X=i) • Joint distribution • P(X1=i1, X2=i2, …, Xn=in) • Continuous case • One variable (pdf) • fX(t) • Joint distribution • fX1, X2,…Xn, (t1,t2,.., tn)

  15. Independent random variables • The random variables X1 , X2 • Are said to be independent if, for all a, b • Example • Green die: X1 • Red die: X2 • X3 = X1 + X2 • X3 and X1 are dependent or independent?

  16. Marginal distribution • Joint distribution • Discrete case • P(X1=i, X2=j), for all i, j in S1xS2 • => • Continuous case • fX1,X2(t1,t2), for all t1,t2 • =>

  17. Expectation of a r.v.: the continuous case • X is a continuous r.v. • Having a probability density function f(x) • The expected value of X is defined by • Define g(X) a function of r.v. X

  18. Expectation of a r.v.: the continuous case (cont’d) • X1, X2, …, Xn: dependent or independent • Example:

  19. Variance, auto-correlation, & covariance • Variance • Continuous case • If are independent r.v. => • If X and Y are correlated r.v.: • Autocorrelation: • Covariance

  20. Conditional probability and conditional expectation: d.r.v. • X and Y are discrete r.v. • Conditional probability mass function • Of X given that Y=y • Conditional expectation of X given that Y=y

  21. Conditional probability and expectation: continuous r.v. • If X and Y have a joint pdf fX,Y(x,y) • Then, the conditional probability density function • Of X given that Y=y • The conditional expectation • Of X given that Y=y

  22. Computing expectations by conditioning • Denote • E[X|Y]: function of the r.v. Y • Whose value at Y=y is E[X|Y=y] • E[X|Y]: is itself a random variable • Property of conditional expectation • if Y is a discrete r.v. • if Y is continuous with density fY(y) => (1) (2) (3)

  23. Proof of equation when X and Y are discrete

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