Random variable X is a mapping that maps each outcome s in the sample space to a unique real number x, x. X s. Real Line

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1 Random Variable Random variable is a mapping that maps each outcome s in the sample space to a unique real number,. s s : outcome Sample Space Real Line Eamples Toss a coin. Define the random variable as follows: 0 if heads if tails Go to the queue at a random time. Define the random variable N as the number of pacets waiting in the queue. Go to the bus stop. Define the random variable W as the time you wait until the bus arrives. Each time we repeat the eperiment, the outcome varies and the value of the random variable varies, according to some probability distribution. With random variables, we can define more complicated things: P e U Y P U, W Rsin, P W w

2 2 Mapping Range (of random variable ) is S { ( s) for some s S} In general, is a many-to-one mapping, but never one-to-many Inverse Operation: For any subset A S, ( A) { s ( s) A} s s : outcome Sample Space Real Line Eample- Bernoulli Toss a biased coin. The coin falls down heads with probability p. Define the random variable as 0 for heads for tails Range: S {0,} P [ ] PHead [ ] p P [ 0.5] 0, P [ 0.] p, P [ 0] p, P [ 2] 0, P [ 3]. is referred to as a Bernoulli random variable. Eample - Geometric Toss a biased coin. is the number of times we toss the coin until we see the first head. Range: S {, 2,3, }. j P [ j] ( p) p for j, 2,3,, where pis the probability of head. is a referred to as a geometric random variable.

3 3 Eample Eponential Random Variable Consider a pacet router that processes arriving pacets. Measure the time between successive pacets arrive, referred to as the pacet inter-arrival times. Let denote the inter-arrival time. Range: S { r 0 r } Eperiments show frequently P [ ] e for 0, where is the pacet arrival rate. is referred to as an eponential random variable. Types of Random Variables Discrete RV Continuous RV Discrete Random Variable The range consists of finite real numbers such as{4,6,8}, or countably infinite real numbers such as{0,, 2, } or {, 2,, 0,, 2, } Notation When we write P, is a random variable, and is a constant or a simple variable. Continuous Random Variable When the range is not countable, the random variable is a continuous one.

4 4 Cumulative Distribution Function The cdf of a random variable is defined as F ( ) P [ ] Eample Uniform Distribution Throw a dart at a spinning wheel. is the phase where the dart hits the wheel. The range of the random variable is S {0 2 }. Within the range, 2 P [ 2] for any phase Therefore the cdf is 2 F ( ) P F 0 2 is a continuous random variable. is referred to as a uniform random variable, or is said to have a uniform probability distribution. In short, U a, b.

5 5 Eample Buses arrive periodically with a period of T. You arrive at the bus stop at a random time. Define random variable W as the time you wait until the net bus arrives. Find the cdf of W. Ans. T F ( ) P 0 T T 0 0 W U(0, T). Eample - Bernoulli Toss a coin. Define as for a head with prob p 0 for a tail with prob p. is referred to as a Bernoulli random variable. F ( ) P[ ] p F p 0 A discrete random variable has discontinuities in its cdf. The value of the cdf is taen approaching from the right.

6 6 Properties of cdf 0 F ( ) 2 F ( ) 3 F ( ) 0 4 F ( ) is a non-decreasing function of 5 F ( ) is continuous from the right: that is, F ( b) lim F ( b h). h0 6 Pa [ b] F ( b) F ( a)

7 7 probability density function (pdf) f ( ) d F ( d ) Properties of pdf Differentiate the cdf to get the pdf. The pdf is the rate at which the cdf f increases. 2 Integrate the pdf to get the cdf: F ( ) f ( u) du 3 To find the probability Pa [ b] F ( b) F ( a) b a f ( u) duu 4 Be careful of the equality sign for discrete random variables. f Eample ( ) 0 for any, andd f ( u) duu. Y is an eponential random variable withh the cdf, F Y ( y ) e i) Find P Y 2. 2 y for y 0. ii) Find the pdf of Y. iii) Plot the cdf and pdf. Ans. i) P Y 2 F ii) f Y y 2 y 2e Y 2 y 0 F Y

8 8 For any pdf, i) nonn - negative ii) area sums to. For any cdf, i) nonn - decreasing from 0 to

9 9 probability mass function The cdf of a discrete random variable has discontinuities. F p 0 The pdf consists of delta functions. f p p 0 When the range is {,,, }, we often use the notation p P[ ]. 0 2 p of a discrete random variable is referred to as the probability mass function (pmf). For a discrete random variable, it is easier to find the pmf first and then the cdf from the following relation F ( ) P[ ] p.

10 0 Notes on continuous distributions. For a continuous random variable, it is easier to find the cdf first and then find the pdf by differentiating the cdf. d f( ) F( ) d 2. It is wrong to say P f ( ) for a continuous random variable. P is always 0 for any with a continuous random variable. However f ( ) may not be zero. Then how are P and f ( ) are related? Note Pa b f ( d ) for any sub-range ab,. Therefore we can state that, for a small, b a P f ( ). () Eq.() is often used for finding the pdf directly.

11 Roots of Popular Random Variables

12 2 Distributions from Tossing Coins Bernoulli Distribution Probability of head in tossing a coin. p is the probability of head: is a Bernoulli random variable. 0 p P p, p P 0 p Geometric Distribution Probability of seeing the first head at the -th toss of a coin. p is the probability of head: is a geometric random variable. p P[ ] ( p) p,,2,3,... Assume 0 p. Eample For any discrete random variable, p must equal. Show p. 0

13 3 Binomial Distribution Probability of heads in n coin tosses: n n p P [ ] p ( p) 0,, 2,,..., n n=20, p= Negative Binomial Distribution Probability of seeing the r-th head at the -th toss: r r p p ( p) p r, r,... r Eample is a negative binomial random variable with r r p p ( p) p r, r,... r. Show p. 0

14 4 Distributions from Random Entry Uniform Distribution U ( ab, ) models a randomly chosen point within a finite interval. pdf: f ( ) a b, a b b a The total area under the pdf should be for all continuous random variables. a b Eample Buses arrive at a bus stop periodically with period of 0 minutes. You arrive at the buss stop at a random time. Let W denote the t time youu wait until the net bus arrives. Then W U(0,0)

15 5 Distributions from Central Limit Theorem Let, 2,, n be a sequence of independent and identically distributed random variables each with mean and variance 2. Define 2 n n Zn. n Then lim Z N(0,) normalized Gaussian distribution. n n 2 Gaussian Distribution N, models a sum of many independent random variables. pdf: 2 ( ) 2 2 f ( ) e 2 2 pdf N(0,) Raleigh Distribution models the amplitude of the sum of two orthogonal, independent gaussian RVs pdf: 2 2b f ( ) e 0 with b 0 b

16 6 Chi-square Distribution ( ) models the sum of squares of independent gaussian RVs. pdf: f ( ) 2 e 0 with, 2, 3, 2 is a special case of Gamma:, and. 2 2 Cauchy Distribution models the ratio of two independent Gaussian RV / pdf: ( ) f 2 2 with 0

17 7 Distributions from Poisson Arrival Process C 0 C C n- C n C n+ t 0 t t n- t n t n+ time Arrivals occur i) in a memoryless manner. ii) P One Arrival during t t o ( t), P No Arrival during t t o0 ( t), P Two or more Arrivals during t o ( t), oj ( t) where oj ( t) is a function such that lim 0. t 0 t We call as the arrival rate, and 2 as the average inter-arrival time. We can show ) probability of arrivals during any time interval of length t is ( t) t P () t e! 2) inter-arrival time is eponential with pdf f ( ) e, 0. Poisson Distribution Probability of arrivals during a time interval t in a poisson arrival process with arrival rate. ( t) t p e 0,, 2,! Eample For all discrete random variables, p. Show p. 0

18 8 Eponential Distribution Ep( ) models the inter-arrival time in a Poisson process with arrival rate. pdf: f ( ) e 0 m-erlang Distribution Erl(, m) models the m-th arrival time in a Poisson process with arrival rate. pdf: m ( ) f ( ) e 0, 0 ( m )! Gamma Distribution G(, ) is a general case of m-erlang. pdf: ( ) f ( ) e 0 with 0, 0 ( ) z ( z) e d Laplacian Distribution is a two-sided eponential 0 Properties are: (), ( z) z( z) z!,. 2 pdf: f ( ) e with 0 2