Random Variables Dr. Abdulaziz Almulhem

Almulhem©20022 Preliminary An important design issue of networking is the ability to model and estimate performance parameters For example, estimate future traffic volumes and characteristics

Almulhem©20023 Why do we need such estimates? To study the effect of routing protocols To estimate resources needed by reservation protocols To study queuing discipline To identify buffer sizes needed

Almulhem©20024 Preliminary Parameters used in characterizing data traffic: Throughput characteristics: 1. Average rate: the load sustained by the source over a time period (resource allocation) 2. Peak rate: the max. load a source can generate (buffering might be needed for smoothing) 3. Variability: burstiness of a source

Almulhem©20025 Preliminary Delay characteristics: 1. Transfer delay: delay from source to destination 2. Delay variation (jitter): variation in transfer delay (impacts real-time applications)

Almulhem©20026 What ’ s next? We need to know little about probability Random variables What are they? Their properties Examples

Almulhem©20027 Probability Premier Probability P(A) of an event A is a number that corresponds to the likelihood that the event A will occur Sample Space (space of events) A B 01 P(A)P(B)

Almulhem©20028 Definitions & observations 0  P(A)  1   P(A i ) = 1; A i is an event in the sample space P(A)= N a /N; N a = number of outcomes in which A occurred (frequency) N= total number of possible outcome

Almulhem©20029 Definitions & observations If two events A & B are mutually exclusive (independent) then: Prob (A or B is to occur) =P(A) + P(B) Prob (A and B to occur) =P(A) * P(B) EX. Out of 2 apples and 3 oranges in a basket, what is the prob. of having 2 oranges when I need to grab three items from the basket?

Almulhem© Definitions & observations The conditional prob. of an event A assuming the event B has occurred P(A|B) is (A & B are not independent): P(A|B)=P(AB)/P(B) If A & B are independent: P(A|B)=P(A) & P(A|B)=P(B)

Almulhem© Baye ’ s Theorem Given the set of mutual exclusive events E 1, …, E n  E i covers an arbitrary event A P(A)=  i n =1 P(A|E i )P(E i )=? Then P(E i |A)=P(A|E i )P(E i )/P(A) E3E3 E1E1 E2E2 A

Almulhem© Example Given, S 0 = event of sending 0 S 1 = event of sending 1 R 0 = event of receiving 0 R 1 = event of receiving 1 P(S 0 ) = p  P(S 1 ) = 1-p Also the received data (bits) can be observed P(R 0 |S 1 ) = p a & P(R 1 |S 0 ) = p b Physical Medium Network SenderReceiver Error

Almulhem© Example Now to calculate the conditional probability of an error That is a one was sent given that a zero is received P(S 1 |R 0 )= P(R 0 |S 1 ) P(S 1 ) / P(R 0 ) Where P(R 0 ) = P(R 0 |S 0 ) P(S 0 ) + P(R 0 |S 1 ) P(S 1 ) P(S 1 |R 0 )=p a p / [p a p+(1-p b )(1-p)]

Almulhem© Random Variables RV is simply a numerical description of the outcome of a random experiment. Examples: Arriving customers at a given time Tossing a coin Packets in a switch at a given time Etc. We describe RV with distribution functions.

Almulhem© Cumulative distribution function (CDF) CDF for an RV denoted F X (x) is defined as the probability that RV is less than or equal to x: F X (x) = p(X  x) F(-  )=0; F(  )=1; 0  F(x)  1 F(x 1 )  F(x 2 ) when x 1  x 2 p(x 1  X  x 2 ) = F(x 2 ) - F(x 1 )

Almulhem© Probability distribution function (pdf) It is the derivative of CDF f X (x) = d F X (x) / dx

Almulhem© Moments To completely characterize a RV, it is sufficient to know its pdf. It is practical to describe some key aspects or few numbers of the pdf rather than specifying the entire pdf. This is called moments or statistical avergares Evaluated using the mathematical expectation

Almulhem© Mathematical expectation The expected or mean value of an RV Expectation is a linear operation

Almulhem© Moments The mth-order moment of F X (x) is Zero order =1 First order is mean (previous slide) Second is the mean-squared value

Almulhem© Moments (cont.) Second moment is the variance and denoted by Standard deviation is square root of variance and it measures the speard of observed values of the RV around its mean

Almulhem© Moments (cont.) Third moment describes the skewness and characterizes the degree of asymmetry of the distribution around its mean. It is a dimensionless quantity. When zero distribution is symmetric +ve leans towards the right; -ve leans towards the left

Almulhem© Moments (cont.) Fourth moments defines kurtosis and measures the flatness or peakedness of a distribution about its mean. It is dimensionless It is relative to the normal distribution More +ve means peaked distribution More – ve means flatten distribution