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2.1 Random Variable Concept

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  1. 2.1 Random Variable Concept • Given an experiment defined by a sample space S with elements s, we assign a real numberto every saccording to some rule as follows; • X : a function that maps all elements of the sample space into points on the real line

  2. Example 2.1-1 • Rolling a die and flipping a coin • Sample space {H,T} x {1,2, ..., 6} • Let the r.v. be a function X as follows • A coin head (H) outcome -> positive value shown up on the die • A coin tail (T) outcome -> negative and twice value shown up on the die

  3. 2.1 Random Variable Concept

  4. Example 2.1-2 • The pointer on a wheel of chance is spun • The possible outcomes are the numbers from 0 to 12 marked on the wheel • Sample space {0 < s 12} • Define a r.v. by the function • Points in S map onto the real line as the set {0 < x 144}

  5. 2.1 Random Variable Concept • Conditions for a Function to be a Random Variable • Every point in S must correspond to only one value of the r.v. • The set { X x} shall be an event for any real number x • The set {X(s)  x} corresponds to those points s in the sample space for which the X(s) does not exceed the number x  This means that the set is not a set of numbers but a set of experimental outcomes • The probabilities of the events {X=} and {X=- } shall be 0: • This condition does not prevent X from being either -  or  for some values of s; it only requires that the prob. of the set of those s be zero The function that maps outcomes in the following manner can not be a r.v. ... ... -  -2 -1 0 1 2

  6. 2.1 Random Variable Concept • Discrete and Continuous Random Variables • Discrete r.v.:Having only discrete values • Example 2.1-1 : discrete r.v. defined on a discrete sample space • Continuous r.v. : Having a continuous range of values • Example 2.1-2 : continuous r.v. defined on a continuous sample space • Mixed r.v. : Some of its values are discrete and some are continuous

  7. 2.2 Distribution Function • Cumulative Probability Distribution Function – CDF • The probability of the event {X x} : • Depend on x • A function of x • Distribution function of X • Properties ; consider a discrete distribution

  8. 2.2 Distribution Function • Discrete Random Variable • If X is a discrete r.v., : stair-step form • Amplitude of a step : probability of occurrence of the value of X where the step occurs • If the values of X are denoted xi, we may write where u(·) is the unit-step function • Using

  9. 2.2 Distribution Function Fx(3)=Fx(3+)=1 by the property (6)

  10. Example 2.2-1 • Let X have the discrete values in the set {-1, -0.5, 0.7, 1.5, 3} • The corresponding probabilities {0.1, 0.2, 0.1, 0.4, 0.2}

  11. Example 2.2-2 • Wheel-of-chance experiment

  12. 2.3 Density Function • Probability Density Function (PDF) of r.v. X • Existence • If the derivative of FX(x) exists, thenfX(x) exists • If there may be places where dFX(x)/dx is not defined (abrupt change in slope), FX(x) is a function with step-type discontinuities such as in ex. 2.2-2.

  13. 2.3 Density Function • Discrete r.v. • Stair-step form distribution function • Description of the derivative of FX(x) at stairsteppoints • Unit-impulse function, (t), used

  14. 2.3 Density Function • Unit-Impulse Function (t) • Definition by its integral property • (x) : any continuous function at the point x = x0 • (t) • Can be integrated as a “function” with infinite amplitude, area of unity, and zero duration • The relationship of unit-impulse and unit-step functions • The general impulse function • Shown symbolically as a vertical arrow occurring at the point x=x0 and having an amplitude equal to the amplitude of the step function for which it is the derivative

  15. 2.3 Density Function • A discrete r.v. • The density function for a discrete r.v. exists • Using impulse functions to describe the derivative of FX(x) at its stair step points

  16. 2.3 Density Function • Properties of Density Functions

  17. Example 2.3-1 • Test the function gX(x) if it can be a valid density function • Property 1 : nonnegative • Property 2: Its area a=1 a=1/

  18. Example 2.3-3 • Find its density function

  19. 2.4 Gaussian r.v. • A r.v. X is called Gaussian if its density function has the form

  20. 2.4 Gaussian r.v. • Numerical or approximation methods for Gaussian r.v. • Tables  many tables according to various • Only one table according to normalized • Consider • For a negative value of x ; • From FX(x), consider

  21. Example 2.4-1 • Find the probability of the event {X5.5} for Gaussian r.v. having aX = 3 and X = 2

  22. Example 2.4-2 • Assume that the height of clouds at some location is Gaussian r.v. X with aX = 1830m and X = 460m • Find the probability that clouds will be higher than 2750m

  23. 2.4 Gaussian r.v. • Evaluations of F(X) by approximation • Ex) 2.4-3 Gaussian r.v. with aX = 7, X = 0.5 Table B-1 : F(0.6)=0.7257

  24. 2.5 Other Distribution and Density • Binomial distribution • Density function & distribution function • Bernoulli trial • For N=6 and p=0.25

  25. 2.5 Other Distribution and Density • Poisson distribution • Density function & distribution function • Quite similar to those for the binomial r.v. • If N and p0 for the binomial case in such a way that Np=b, the Poisson case results • Applications • The number of defective units in sample taken from a production line • The number of telephone calls made during a period of time • The number of electrons emitted from a small section of cathode in a given time interval • If the time interval of interest has duration T, and the events being counted are known to occur at an average rate  and have a Poisson distribution, then b = T.

  26. 2.5 Other Distribution and Density • Uniform distribution • The quantization of signal samples prior to encoding in digital communication systems  The error introduced in the round-off process  uniform distributed • Figure 2.5-2

  27. 2.5 Other Distribution and Density • Exponential

  28. 2.5 Other Distribution and Density • Rayleigh • The envelope of one type of noise when passed through a bandpass filter

  29. 2.6 Conditional Distribution and Density Functions • Conditional Probability • For two events A and B where P(B)0, the conditional probability of A given B • Conditional Distribution • A : identified as the event {X  x} for the r.v. x • Conditional distribution function of X • = the joint event • This joint event consists of all outcomes s such that • The conditional distribution : discrete, continuous, or mixed random variables

  30. 2.6 Conditional Distribution and Density Functions • Properties of Conditional Distribution

  31. 2.6 Conditional Distribution and Density Functions • Conditional Density • Conditional density function of the r.v. x • The derivative of the conditional distribution function • If FX(x|B) contains discontinuities, impulse response are present in fX(x|B) to account for the derivatives at the discontinuities • Properties

  32. 2.6 Conditional Distribution and Density Functions • Example 2.6-1 • Red, green, and blue balls in two boxes as shown in the following table • Select a box first, and then take a ball from the selected box • The 2nd box is larger than the 1st box  causing the 2nd box to be selected more frequently • Event selecting the 2nd box : B2, P(B2)=8/10 • Event selecting the 1st box : B1, P(B1)=2/10 • Event selecting a red ball : r.v. x1 • Event selecting a green ball : r.v. x2 • Event selecting a blue ball : r.v. x3

  33. 2.6 Conditional Distribution and Density Functions

  34. 2.6 Conditional Distribution and Density Functions • total probability

  35. 2.6 Conditional Distribution and Density Functions • Event B : Case 1. bx : Case 2. b>x :

  36. 2.6 Conditional Distribution and Density Functions • Conditional distribution • From the assumption that the conditioning event has nonzero prob., • Similarly,

  37. 2.6 Conditional Distribution and Density Functions • Example 2.6-2 • Sky divers try to land within a target circle • Miss-distance from the point has the Rayleigh distribution withb=800m2 and a=0 • Target : a circle of 50m radius with a bull’s eye of 10m radius • The prob. of sky diver hitting the bull’s eye, given that the landing is on the target ? Sol) x = 10, b = 50

  38. ta tb 0 T Random Poisson Points • Papoulis pp. 117 • Points in nonoverlapping intervals • P{ka in ta, kb in tb} where A = {ka in ta}, B = {kb in tb}

  39. Random Poisson Points

  40. Homework • Prob. 2.3-1, 2.3-5, 2.3-9, 2.3-12, 2.3-13, 2.4-3, 2.4-4, 2.5-7 • Due : next Tuesday