Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

Random variables (r.v.) Random variable Definition: a variable that takes on various values in a random way Examples Number of items to check out at super-market Discrete (takes on a countable number of values) End-to-end delay in a communication network Continuous ([propagation delay; +infinity]) Described thru Probability mass function (pmf) Probability density function (pdf) 1 Probability distribution functions The behavior of a Discrete random variable is captured by a probability mass function Example: # items to check out at super-market (X) P[X = i] and i = 1, 2, . . ., 200 (a certain upper bound) of a continuous random variable is captured by a probability density function Example: End-to-end delay in a communication network (D) fD(d) 2 Probability theory Outcome Origin Experiments 1/2 H Example 1 1/2 Toss a coin 3 times H 1/2 1/2 H HHH 1/2 T HHT 1/2 H HTH 1/2 T HTT H THH T THT H TTH T TTT 1/2 Possible r.v. X = # of heads T T 1/2 1/2 H 1/2 1/2 1/2 T A discrete r.v. Sample space 1/2 3 Example 1 continued Sample space X {0,1,2,3} What is the probability distribution of X? Building on the following observation P(X = i) = # of feasible sample points / total # of sample points P(X=0) = 1/8; P(X=1) = 3/8 P(X=2) = 3/8; P(X=3) = 1/8 P( X i) 1 4 Example 2: cost functions Cost function X = cost_function (sample points) Example If heads appears you make 10 $ If tail appears you pay 5 $ X = amount of money you make => X = {-15, 0, 15, 30} (discrete r.v.) 5 Example 3: continuous r.v. End to end delay in a communication network X Continuous random variable belongs to a state space Lower bound = propagation delay Upper bound = +infinity 6 Discrete random variables: probability mass functions Discrete random variables Classified according to their probability mass function I will cover Binomial distribution Geometric distribution Poisson distribution 7 Binomial distribution Binomial distribution is primarily associated with the tossing of a coin A certain number of independent trials Outcome #1 (or 1) with probability p (referred to as success) Outcome#2 (or 2) with probability 1-p (referred to as failure) Example: n trials (n=6) What is the probability that the following sequence arises? 1, 2, 1, 2, 1, 2 Answer: Prob = p (1-p) p (1-p) p (1-p) = p3(1-p)3 8 Binomial random variable Suppose n independent trials resulting in a “success” with probability p And in a “failure” with probability (1-p) If X represents the number of successes in the n trials => X is a binomial random variable with parameters (n, p) 9 Binomial distribution: example1 Example: Four fair coins are flipped. What is the probability that two heads and two tails are obtained? Solution Let X equal the number of heads (successes) => X is a binomial r.v. with parameter (n=4, p=1/2) 4 1 P( X 2) 2 2 2 2 1 3 2 8 10 Binomial distribution: example2 Example It is known that any item produced by a certain machine will be defective with probability 0.1, independent of any other item. What is the probability that in A sample of three items At most one will be defective? 11 Geometric random variable Experiment n trials Each having probability p of being a success Are performed until a success occurs If X is the number of trials required until the first success X is a geometric r.v. with parameter p Its probability mass function is 12 Geometric r.v.: application Time sharing Jobs running on a computer X: Represents how many times a job cycles around gets queued in order to use the CPU A quantum of time is assigned to each process => is a geometric r.v. 6 tosses of a coin The first outcome is Heads How many more heads do I need before I get a tail? 13 Poisson distribution Poisson distribution is Associated with the observation of event occurrences T=15 min 0 Event#1 Time If N represents the number of events in T => N/T = average number of events /minute interested in answering the following question How many occurrences of this event take place per minute? The way it has been done Either 0 or 1 event occurrence per minute 14 Poisson distributed random variable A Poisson random variable X Characterizes the number of occurrences of an event Typically an arrival => X = # arrivals per unit time With parameter λ (average # of arrivals per unit time) p( X i ) e i i! The value of λ (arrival rate) # arrivals ( N ) lim T T 15 Poisson distribution: example 1 Example If number of accidents occurring on a highway per day is a Poisson r.v. with parameter λ = 3, What is the probability that no accidents occur today? Solution P( X 0) e3 0.05 16 Poisson distribution: example 2 Consider an experiment that counts the number of α-particles emitted in a one-second interval by one gram of radioactive material. If we know that ,on average , 3.2 such α-particles are given off what is a good approximation to the probability that no more than 2 α-particles appear? P{ X 2} e 3.2 3.2e 3.2 (3.2) 2 3.2 e 0.382 2 17 Binomial approximation to the Poisson distribution 0 N events ΔT Time Divide the time axis into ΔT small enough so that T At most only one arrival can occur n ΔTs are required It is like creating a binomial experiment Each ΔT is a trial Outcome: 0 arrivals (p ?) or 1 arrival ((1-p)?) 18 Binomial approximation to the Poisson distribution (cont’d) With what probability we are going to have 1 arrival or 0 arrivals in one ΔT ? Average arrival rate per ΔT interval N .T 0 p 1 (1 p) T As such N T Pr(0arrivals ) 1 T N T Pr(1arrivals ) T i n i n T .N T .N Pr( X i) 1 T i T 19 Binomial approximation to the Poisson distribution (cont’d) n T .N T .N Pr( X i ) 1 T i T 1 N T ; n T i n i If you let n tends to infinity you will get Pr( X i ) e i i! 20 Cumulative distribution Consider a discrete r.v. X Taking on the values from 0 to infinity The cumulative distribution function can be expressed F (j) = P(X <= j) = P(X=0) + .... + P(X=j-1) + P(X=j) For instance Suppose X has a probability mass function given by P(1) = ½, P(2) = 1/3, P(3) = 1/6 The cumulative function F of X is given by 0, j 1 1 ,1 j 2 2 F ( j) 5 ,2 j 3 6 21 1,3 j Residual distribution Given by P( X j ) P( X i) i j P( X i) 1 P( X j ) 1 P( X j ) i 0 22 Expectation of a discrete random variable X is a discrete random variable Having a probability mass function p(X) => Expected value of X is defined by E[X] as E[ X ] x. p( x) x E[aX b] aE[ X ] b a and b are constants Variance of X Var ( X ) E[ X 2 ] ( E[ X ]) 2 23 Expectation: example 1 Find E[X] where X is the outcome when we roll a fair dice 1 1 1 1 1 1 7 E[ X ] 1 2 3 4 5 6 6 6 6 6 6 6 2 Find Var(X) when X represent the outcome when we roll a fair dice 1 1 1 1 1 1 91 E[ X ] 1 4 9 16 25 36 6 6 6 6 6 6 6 2 2 91 7 35 Var ( X ) 6 2 12 24 Expectation: example 2 Calculate E[X] when X is Binomially distributed With parameters n and p n i E[ X ] ip (i ) i p (1 p ) n i i 0 i 0 i n n in! n! i n i p (1 p) p i (1 p ) n i i 1 ( n i )!i! i 1 ( n i )! (i 1)! n n n 1 n 1 k n! i 1 n i p (1 p ) n 1 k np p (1 p) np i 1 ( n i )! (i 1)! k 0 k n np[ p (1 p )] n 1 np 25 Expectation: example 3 Find E[X] Of a geometric random variable X with parameter p E[ X ] np (1 p ) n 1 n 1 p nq n 1 n 1 d n d n E[ X ] p q p q dq n 1 n 1 dq p d q p 1 dq 1 q 1 q 2 p 26 Expectation: example 4 Calculate E[X] For Poisson random variable X with parameter λ ie i e i E[ X ] i! i 0 i 1 (i 1)! e i 1 k (i 1)! e k! e e . i 1 k 0 as : k 0 k / k! e . 27