Random Variables and Cumulative Distribution


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1 Probbility: One Rndom Vrible 3 Rndom Vribles nd Cumultive Distribution A probbility distribution shows the probbilities observed in n experiment. The quntity observed in given tril of n experiment is number clled rndom vrible (RV). In the following, RVs re designted by boldfce letters such s x nd y. Discrete RV: vrible tht cn only tke on certin discrete vlues. Continuous RV: vrible tht cn ssume ny vlue within specified rnge (possibly infinite). For given RV x, there re three primry events to consider involving probbilities: {x }, { < x b}, {x > b} For the generl event {x x}, where x is ny rel number, we define the cumultive distribution function (CDF) s F x (x) = Pr(x x), < x < The CDF is probbility nd thus stisfies the following properties: 1. 0 F x (x) 1, < x < 2. F x () F x (b), for < b 3. F x ( ) = 0, F x ( ) = 1 We lso note tht Pr( < x b) = F x (b) F x () Pr(x > x) = 1 F x (x)
2 Probbility: One Rndom Vrible 15 Functions of One RV In mny cses, n exmintion is necessry of wht hppens to RV x s it psses through vrious trnsformtions, such s rndom signl pssing through nonliner device. Suppose tht the output of some nonliner device with input x cn be represented by the new RV: y = g(x) If the PDF of x is known to be f x (x), nd the function y = g(x) hs unique inverse, the PDF of y is relted by f y (y) = f x(x) g (x) If the inverse of y = g(x) is not unique, nd x 1, x 2,..., x n re ll of the vlues for which y = g(x 1 ) = g(x 2 ) = = g(x n ), then the previous reltion is modified to f y (y) = f x(x 1 ) g (x 1 ) + f x(x 1 ) g (x 1 ) + + f x(x n ) g (x n ) Another method for finding the PDF of y involves the chrcteristic function. For exmple, given tht y = g(x), the chrcteristic function for y cn be found directly from the PDF for x through the expected vlue reltion Φ y (s) = E[e isg(x) ] = e isg(x) f x (x)dx Consequently, the PDF for y cn be recovered from chrcteristic function Φ y (s) through inverse reltion f y (y) = 1 e isy Φ y (s)ds 2π
3 16 Probbility: One Rndom Vrible Exmple: SqureLw Device The output of squrelw device is defined by the qudrtic trnsformtion y = x 2, > 0 where x is the RV input. Find n expression for the PDF f y (y) given tht we know f x (x). Solution: We first observe tht if y < 0, then y = x 2 hs no rel solutions; hence, it follows tht f y (y) = 0 for y < 0. For y > 0, there re two solutions to y = x 2, given by x 1 =, x 2 = where g (x 1 ) = 2x 1 = 2 y g (x 2 ) = 2x 2 = 2 y In this cse, we deduce tht the PDF for RV y is defined by f y (y) = 1 [ ( ) ( y 2 f x + f x y )] U(y) where U( y) is the unit step function. It cn lso be shown tht the CDF for y is [ ( ) ( y F y (y) = F x F x )] U(y)
4 54 Rndom Processes Exmple: Correltion nd PDF Consider the rndom process x(t) = cosωt + bsinωt, where ω is constnt nd nd b re sttisticlly independent Gussin RVs, stisfying Determine = b = 0, 2 = b 2 = σ 2 1. the correltion function for x(t), nd 2. the secondorder PDF for x 1 nd x 2. Solution: (1) Becuse nd b re sttisticlly independent RVs, it follows tht b = b = 0, nd thus or R x (t 1, t 2 ) = (cosωt 1 +bsinωt 1 )(cosωt 2 +bsinωt 2 ) = 2 cosωt 1 cosωt 2 + b 2 sinωt 1 sinωt 2 = σ 2 cos[ω(t 2 t 1 )] R x (t 1, t 2 ) = σ 2 cosωτ, τ = t 2 t 1 (2) The expected vlue of the rndom process x(t) is x(t) = cosωt + b sinωt = 0. Hence, σ 2 x = R x(0) = σ 2, nd the firstorder PDF of x(t) is given by f x (x, t) = 1 σ /2σ2 e x2 2π The secondorder PDF depends on the correltion coefficient between x 1 nd x 2, which, becuse the men is zero, cn be clculted from nd consequently, f x (x 1, t 1 ; x 2, t 2 ) = ρ x (τ) = R x(τ) R x (0) = cosωτ 1 2πσ 2 sinωτ exp ( x2 1 2x 1x 2 cosωτ + x 2 2 2σ 2 sin 2 ωτ )
5 Trnsformtions of Rndom Processes 73 Memoryless Nonliner Trnsformtions Consider system in which the output y(t 1 ) t time t 1 depends only on the input x(t 1 ) nd not on ny other pst or future vlues of x(t). If the system is designted by the reltion y(t) = g[x(t)] where y = g(x) is function ssigning unique vlue of y to ech vlue of x, it is sid tht the system effects memoryless trnsformtion. Becuse the function g(x) does not depend explicitly on time t, it cn lso be sid tht the system is time invrint. For exmple, if g(x) is not function of time t, it follows tht the output of time invrint system to the input x(t +ε) cn be expressed s y(t +ε) = g[x(t +ε)] If input nd output re both smpled t times t 1, t 2,..., t n to produce the smples x 1,x 2,...,x n nd y 1,y 2,...,y n, respectively, then y k = g(x k ), k = 1,2,..., n This reltion is trnsformtion of the RVs x 1,x 2,...,x n into new set of RVs y 1,y 2,...,y n. It then follows tht the joint density of the RVs y 1,y 2,...,y n cn be found directly from the corresponding density of the RVs x 1,x 2,...,x n through the bove reltionship. Memoryless processes or fields hve no memory of other events in loction or time. In probbility nd sttistics, memorylessness is property of certin probbility distributions the exponentil distributions of nonnegtive rel numbers nd the geometric distributions of nonnegtive integers. Tht is, these distributions re derived from Poisson sttistics nd s such re the only memoryless probbility distributions.
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