Download - Probability, Random Variables and Random Processes - Part 2

8/8/2019 Probability, Random Variables and Random Processes - Part 2

1/43


2/43


3/43


4/43

1. Can an arbitrary mapping be a Random Variable? (P(X


5/43

1. RV is a function (single valued) whose domain is the sample space and whoserange is the real line.

2. Random Variable: Outcome of the experiment is random; the value an RV takesis a Variable; hence the name Random Variable

3. When are two RVs said to be equivalent? ( iff P[ { w: x1(w) not equals x2(w) } ]= 0 )


6/43

1. Mixed case is least important type of RV, but it occurs in some problems ofPractical Interest.

E.g.; Spin a Pointer on the Wheel with numbers from 0 to 99 and denote an R.VX= 0 when pointer stops anywhere between 0 and 10 and X= 1 when pointer

stops anywhere between 11 and 20 and X= the pointer value itself when stopsanywhere else.

Note: Mixed RV will have a discrete but not a stair-case CDF

1. What are/can be the Sample Spaces (Continuous or Discrete) for each of theabove three types of RVs?

2. These can similarly be classified based on their CDF also.

3. So, enough of this, but how do we express an RV?


7/43


8/43

1. In fact, if you go on drawing a graph while constrained by the above threeconditions in the slide, that would eventually become a valid Distributionfunction.

2. Additional properties?

3. P{X > x} = 1 F(x)

4. P{x1x1

5. P{X = x} = F(x) F(x -) Eq 4

6. At a discontinuity point of the distribution, the left and right limits are differentand so, from above Eq 4, P{X = x0} = F(x0) F(x0 -) > 0 Eq 5 and thus theonly discontinuities can be of the jump type and occur at points x0 where Eq 5 issatisfied. These points always can be enumerated in sequence and they are atmost countable in number.


9/43

1. While the CDF introduced in the last section represents a mathematical tool tostatistically describe a random variable, it is often quite cumbersome to workwith CDFs. For example, we will see later in this talk that the most important andcommonly used random variable, the Gaussian random variable, has a CDF that

cannot be expressed in closed form. Furthermore, it can often be difficult to infervarious properties of a random variable from its CDF. To help circumvent theseproblems, an alternative and often more convenient description known as theprobability density function(PDF) is often used.

2. The probability density function (PDF) of the r.v. Xevaluated at the point xis theprobability that the r.v. Xlies in an infinitesimal interval about the point X= x,normalized by the length of the interval.

3. Non-negativity property of the pdf follows from the monotonically increasingnature of the CDF.


10/43

1. What can you get from a given pdf?

2. pmf because it looks like point wise and like probability, mass also is conserved.

3. Properties of pdf: 1.non negative 2.total area unity, etc.


11/43

1. Gaussian : First used by De Moviere for modeling measurement errors and laterjustified by Laplace and later rigorously justified by Carl F. Gauss. Lot of otherdistributions (e.g., binomial and poisson) can be approximated by this.


12/43


13/43


14/43


15/43


16/43


17/43

1. How to generate a Gaussian distribution given an uniform distribution[0,1)?BoxMuller Method:


18/43

1. Note that rho-squared in the Rayleigh case is not its variance.


19/43

1. fXY(u,v) is the joint density function of RVs X and Y.

2. It is easy to show that the above fX|Y (x|Y= y) satisfies the required propertiesof a density function. Its indeed a valid pdf.

3. Similarly the equation for fY|X (y|X= x)


20/43

1. Implying that the conditional p.d.fs coincide with their unconditional p.d.fs. Thismakes sense, since if Xand Yare independent r.vs, information about Y

shouldnt be of any help in updating our knowledge about X.

2. We can easily prove Bayes Theorem for p.d.fs from the previous ideas

discussed.


21/43

1. fXY(u,v) is the joint density function of RVs X and Y.

2. It is easy to show that the above fX|Y (x|Y= y) satisfies the required propertiesof a density function. Its indeed a valid pdf.

3. Similarly the equation for fY|X (y|X= x)


22/43


23/43


24/43


25/43


26/43


27/43


28/43

1. For a Gaussian rv (with mean=0) X, Y = X2 is a Chi-Square rv with one degreeof freedom (n = 1).


29/43


30/43

*** For references, plz refer the last slideof Part-1 of this series. ***


31/43


32/43


33/43


34/43


35/43


36/43


37/43


38/43

1. Consider Y = a + X. |X|


39/43


40/43


41/43


42/43


43/43

Download - Probability, Random Variables and Random Processes - Part 2

Top Related