ECE 302: Lecture 4.7 Gaussian Random Variable

1 Stanley Chan 2022. All Rights 302: Lecture Gaussian Random VariableProf Stanley ChanSchool of Electrical and Computer EngineeringPurdue University1 / 22 Stanley Chan 2022. All Rights schedule:Continuous Random variables, PDFCDFE xpectationMean, mode, medianCommon Random variablesUniformExponentialGaussianTrans formation of Random variablesHow to generate Random numbersToday s Lecture :Definition of GaussianMean and varianceSkewness and kurtosisOrigin of Gaussian2 / 22 Stanley Chan 2022. All Rights an Gaussian Random Variable . The PDF ofXisfX(x) =1 2 2e (x )22 2(1)where ( , 2) are parameters of the distribution. We writeX Gaussian ( , 2)orX N( , 2)to say thatXis drawn from a Gaussian distribution of parameter ( , 2).3 / 22 Stanley Chan 2022. All Rights the mean and = -3 = = 0 = = = = 1 = 2 = 3 = 4 changes, = 1 = 0, changesFigure: A Gaussian Random Variable with different and 4 / 22 Stanley Chan 2022.

2 All Rights the meanTheoremIfX N( , 2), thenE[X] = ,andVar[X] = 2.(2)E[X] =1 2 2Z xe (x )22 2dx(a)=1 2 2Z (y+ )e y22 2dy=(b)=(c)= .5 / 22 Stanley Chan 2022. All Rights the varianceTheoremIfX N( , 2), thenE[X] = ,andVar[X] = 2.(3)Var[X] =1 2 2Z (x )2e (x )22 2dx(a)= 2 2 Z y2e y22dy,by lettingy== 2 2 ye y22 + 2 2 Z e y22dy== 26 / 22 Stanley Chan 2022. All Rights Gaussian PDFD efinitionAstandard Gaussian (or standard Normal) Random variableXhas a PDFfX(x) =1 2 e x22.(4)That is,X N(0,1) is a Gaussian with = 0 and 2= : Definition of the CDF of the standard Gaussian (x).7 / 22 Stanley Chan 2022. All Rights Gaussian CDFD efinitionTheCDFof the standard Gaussian is defined as the ( ) function (x)def=FX(x) =1 2 Zx e t22dt.(5)The standard Gaussian s CDF is related to a so-callederror functionwhich is defined aserf(x) =2 Zx0e t2dt.

3 (6)It is quite easy to link (x) with erf(x): (x) =12 1 + erf x 2 ,anderf(x) = 2 (x 2) / 22 Stanley Chan 2022. All Rights of arbitrary GaussianTheorem (CDF of an arbitrary Gaussian )LetX N( , 2). Then,FX(x) = x .(7)We start by expressingFX(x):FX(x) =.Substitutingy=t , and using the definition of standard Gaussian , wehaveZx 1 2 2e (t )22 2dt=Zx 1 2 e y22dy=9 / 22 Stanley Chan 2022. All Rights resultsP[a<X b] = b a .(8)To see this, note thatP[a<X b] =P[X b] P[X a] = b a .CorollaryLetX N( , 2). Then, the following results hold: (y) = 1 ( y).P[X b] = 1 b .P[|X| b] = 1 b + b 10 / 22 Stanley Chan 2022. All Rights and KurtosisDefinitionFor a Random variableXwith PDFfX(x), define the followingcentralmomentsasmean =E[X]def= ,variance =Eh(X )2idef= 2,skewness =E" X 3#def= ,kurtosis =E" X 4#def=.

4 11 / 22 Stanley Chan 2022. All Rights is skewness?E X 3 .Measures howasymmetricalthe distribution has skewness skewnesssymmetricnegative skewnessFigure: Skewness of a distribution measures how asymmetric the distribution this example, the skewness are: orange = , black = 0, blue = / 22 Stanley Chan 2022. All Rights is kurtosis? =E X 4 .Measures howheavy tailis. Gaussian has kurtosis people preferexcess kurtosis 3. Gaussian has excesskurtosis > 0kurtosis = 0kurtosis < 0 Figure: Kurtosis of a distribution measures how heavy tail the distribution is. Inthis example, the (excess) kurtosis are: orange = , black = 0, blue = / 22 Stanley Chan 2022. All Rights and KurtosisRandom variableMeanVarianceSkewnessExcess kurtosis 2 3 Bernoullipp(1 p)1 2p p(1 p)11 p+1p 6 Binomialnpnp(1 p)1 2p np(1 p)6p2 6p+1np(1 p)Geometric1p1 pp22 p 1 pp2 6p+61 pPoisson 1 1 Uniforma+b2(b a)2120 65 Exponential1 1 226 Gaussian 200 Table: The first few moments of commonly used Random / 22 Stanley Chan 2022.

5 All Rights : TitanicOn April 15, 1912, rms titanic sank after hitting an iceberg. This haskilled 1502 out of 2224 passengers and crew. A hundred years later, wewant to analyze the data. is a dataset collecting the identities, age, gender, etc of 1 (Died)Group 2 (Survived) / 22 Stanley Chan 2022. All Rights : TitanicMean and standard deviation cannot tell the and kurtosis can tell the 1 (died)Group 2 (survived)Figure: The titanic / 22 Stanley Chan 2022. All Rights of GaussianWhere does Gaussian come from?Why are they so popular?Why do they have bell shapes?What is the origin of Gaussian ?When wesummany independent Random variables, the resultingrandom Variable is a is known as theCentral Limit Theorem. The theoremapplies toanyrandom Random variables is equivalent toconvolvingthePDFs. Convolving PDFs infinitely many times yields the / 22 Stanley Chan 2022.

6 All Rights experiment of throwing many dices(a)X1(b)X1+X2(c)X1+..+X5(d)X1+..+X1 00 Figure: When adding uniform Random variables, the overall distribution isbecoming like a / 22 Stanley Chan 2022. All Rights ofXandY= Convolution offXandfYExample: Two rectangles to give a triangle:We will show this result in a later Lecture :(fX fX)(x) =Z fX( )fX(x )d .19 / 22 Stanley Chan 2022. All Rights you convolve infinitely many timesThen in Fourier domain you will haveF{(fX fX .. fX)}=F{fX} F{fX} .. F{fX}. (sin x)/x(sin x)2/x2(sin x)3/x3 Figure: Convolving the PDF of a uniform distribution is equivalent to multiplyingtheir Fourier transforms in the Fourier space. As the number of convolutiongrows, the product is gradually becoming / 22 Stanley Chan 2022. All Rights of GaussianWhat happens if you convolve a PDF infinitely many times?You will get a is known as the central limit are Gaussians everywhere?

7 We seldom look at individual Random variables. We often lookat the we have a sum,Central Limit Theoremkicks Random variables is equivalent toconvolvingthePDFs. Convolving PDFs infinitely many times yields the result applies toanyrandom Variable , as long as they areindependently / 22 Stanley Chan 2022. All Rights / 22