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+{
+"cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Chapter 2: PROBABILITY RANDOM SIGNALS AND RANDOM PROCESS"
+ ]
+ },
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.12: Probability_density_function.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Probability density function \n",
+"//Example 2.12\n",
+"//Find find the value of k\n",
+"clc;\n",
+"clear;\n",
+"function y=FX1(x) //for -infinte<x<=0\n",
+" y=0 \n",
+"endfunction\n",
+"function y=FX2(x) //for 0<x<=10\n",
+" y=k*x^2 \n",
+"endfunction\n",
+"function y=FX3(x) //for 10<x<infine\n",
+" y=100*k\n",
+"endfunction\n",
+"k=poly(0,'k');\n",
+" //from the expression for CDF is given\n",
+"y=100*k //for 10<x<infine\n",
+"y==1; \n",
+"k=1/100; //k=y/100\n",
+"disp(k,'i) k = ');\n",
+"//CDF function can be expressed\n",
+"// FX(x)=P(X<=x)\n",
+"P5=FX2(5); //x=5\n",
+"disp(P5,'ii) P(X<=5) = ');\n",
+" //now differentiating with respect tox we ,have \n",
+" //PDF fX=0 for -infinte<x<=0,10<x<infine\n",
+"x=poly(0,'x');\n",
+"m=x^2/100;\n",
+"df=derivat(m); // for 0<x<=10\n",
+"disp(' for 0<x<=10',df,'iii)PDF a) fX(x) =');\n",
+"disp(' -infinte<x<=0,10<x<infine',0,' b)fX(x) = ');\n",
+"\n",
+"x1=5,x2=7;\n",
+"function y=z(x),\n",
+" y=x/50;\n",
+"endfunction\n",
+"P=intg(x1,x2,z);\n",
+"disp(P,'iv) P(5<X<=7) =');"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.13: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Probability \n",
+"//Example 2.13\n",
+"//page no 54\n",
+"//find the probability \n",
+"clc;\n",
+"clear;\n",
+"function y=f(x),\n",
+" y=2*exp(-(2*x)),//Probability Density Function\n",
+"endfunction\n",
+"\n",
+"a=1;\n",
+"b=3;\n",
+"P=intg(a,b,f);//probability that random variable between 1 and 3\n",
+"disp(P,'P(1<X<3)=');"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.14: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Probability \n",
+"//Example 2.14\n",
+"//page no 54\n",
+"//find the probability \n",
+"clc;\n",
+"clear;\n",
+"function y=f(x),\n",
+" y=12*x^3-21*x^2+10*x,//Probability Density Function\n",
+"endfunction\n",
+"a=0;\n",
+"b=1/2;\n",
+"P=intg(a,b,f);\n",
+"disp(P,'P(X<=1/2)=');\n",
+"disp(1-P,'P(X>1/2)=');"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.1_a: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Probability\n",
+"//Example 2.1.a\n",
+"//page no 43\n",
+"//Find the Probability \n",
+"clc;\n",
+"clear;\n",
+"redballs=3;\n",
+"whiteballs=4;\n",
+"blackballs=5; \n",
+"//P=ways of choosing a red ball/Total no. of ways choosing a ball\n",
+"Pro=redballs/(redballs+whiteballs+blackballs);\n",
+"disp('Probability Getting red ball is');\n",
+"disp(Pro,'P(R)=');\n",
+""
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.1_b: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"\n",
+"//Caption: Probability\n",
+"//Example 2_1b\n",
+"//page no 43\n",
+"//Find the probability Getting black ball\n",
+"clc;\n",
+"clear;\n",
+"redballs=3;\n",
+"whiteballs=4;\n",
+"blackballs=5;\n",
+"//Probability Getting black ball\n",
+"prob=blackballs/(redballs+whiteballs+blackballs);\n",
+"p=1-prob;\n",
+"disp('Probability not getting black ball');\n",
+"disp(p,'P(B~)=');"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.1_c: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Probability\n",
+"//Example 2.1.c\n",
+"//page no 43\n",
+"//Find the probability \n",
+"clc;\n",
+"clear;\n",
+"redballs=3;\n",
+"whiteballs=4;\n",
+"blackballs=5;\n",
+"//Probability Getting black ball\n",
+"pb=blackballs/(redballs+whiteballs+blackballs);\n",
+"//Probability Getting white ball\n",
+"pw=whiteballs/(redballs+whiteballs+blackballs);\n",
+"//Probability Getting white ball or black ball\n",
+"pwb=pb+pw// black and white are mutuallly exclusive events\n",
+"disp('Probability Getting white ball or black ball will be ')\n",
+"disp(pwb,'P(B+W)=')\n",
+""
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.20: Joint_probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Joint probability\n",
+"//Example 2.20\n",
+"//Page no 60\n",
+"//Find Determine the value of constant C\n",
+"clc;\n",
+"clear;\n",
+"X=[0,0;2,2;2,0];\n",
+"Y=[0,0;3,3;0,3];\n",
+"function y=f(x,y),\n",
+" y=2*x+y;\n",
+"endfunction\n",
+"P=int2d(X,Y,f);\n",
+"C=poly(0,'C');\n",
+"//we know that joint PDF \n",
+"// double integration fXY(x,y)dxdy=1\n",
+"//C*P==1;\n",
+"C=1/P;\n",
+"disp(C,'C = ')"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.23: probability_density_function.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: probability density function \n",
+"//Example 2.23\n",
+"//page no \n",
+"//verify the area under curve is unity,prove that the mean is zero\n",
+"clc;\n",
+"clear;\n",
+"//continuous random variable X in the range(-3,3)\n",
+"//PDF of fucntions\n",
+"function y=fX1(x)\n",
+" y=((3+x)^2)/16; //for -3<=x<=-1\n",
+"endfunction\n",
+"a1=-3;\n",
+"a2=-1;\n",
+"\n",
+"function y=fX2(x), //for-1<=x<=1\n",
+" y=(6-2*x^2)/16;\n",
+"endfunction\n",
+"b1=-1;\n",
+"b2=1;\n",
+"\n",
+"function y=fX3(x)\n",
+" y=((3-x)^2)/16; //for 1<=x<=3\n",
+"endfunction\n",
+"c1=1;\n",
+"c2=3;\n",
+"//area under pdf curve\n",
+"//area = integral Fx(x)dx from -3 to 3\n",
+"area =intg(a1,a2,fX1)+intg(b1,b2,fX2)+intg(c1,c2,fX3);\n",
+"disp(area,'Area =');\n",
+"disp(' Therefore area under PDF curve is unity.');\n",
+"function y=f1(x)\n",
+" y=(x*(3+x)^2)/16; //for -3<=x<=-1\n",
+"endfunction\n",
+"a1=-3;\n",
+"a2=-1;\n",
+"\n",
+"function y=f2(x), //for-1<=x<=1\n",
+" y=(x*(6-2*x^2))/16;\n",
+"endfunction\n",
+"b1=-1;\n",
+"b2=1;\n",
+"function y=f3(x)\n",
+" y=(x*(3-x)^2)/16; //for 1<=x<=3\n",
+"endfunction\n",
+"c1=1;\n",
+"c2=3;\n",
+"mx=intg(a1,a2,f1)+intg(b1,b2,f2)+intg(c1,c2,f3);//mean value\n",
+"disp(mx,'Mean value =');\n",
+"disp(' Therefore mean value is zero.');\n",
+"//Therefore area under PDF curve is unity\n",
+"//Hence Therefore mean value is zero"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.24: mean_mean_Square_Standard_deviation.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: mean ,mean Square Standard deviation\n",
+"//Example 2.24\n",
+"//page no \n",
+"//Find mean,mean Square ,Standard deviation\n",
+"clc;\n",
+"clear;\n",
+"function y=f(x)\n",
+" y=x/(2*%pi); //for 0<=x<=2*%pi\n",
+"endfunction\n",
+"\n",
+"a=0;\n",
+"b=2*%pi;\n",
+"mx=intg(a,b,f);//mean value\n",
+"disp(mx,'Mean value mx=');\n",
+"\n",
+"function y=fX(x)\n",
+" y=x^2/(2*%pi); //for 0<=x<=2*%pi\n",
+"endfunction\n",
+"a=0;\n",
+"b=2*%pi;\n",
+"//X^2=E(X^2) \n",
+"X2=intg(a,b,fX);//mean square value\n",
+"disp(X2,'Mean square value X^2 =');\n",
+"\n",
+"sigma2=X2-mx^2;//variance\n",
+"sigma=sqrt(sigma2);//Standard deviation\n",
+"disp(sigma,'Standard deviation sigma=')"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.2: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Probability\n",
+"//Example 2.2\n",
+"//page no 43\n",
+"//find the probability of getting 5\n",
+"clc;\n",
+"clear;\n",
+"total_possibleoutcomes=6*6;\n",
+"probabilityofeachoutcome=1/total_possibleoutcomes;//probability of each outcome\n",
+"noofways=4; //ways of getting 5\n",
+"probability=noofways*probabilityofeachoutcome;//probability of getting 5\n",
+"disp(probability,'Probability of getting 5 is');\n",
+""
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.31: Mean_and_Variance.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Mean,Variance\n",
+"//Example 2.31\n",
+"//page no 85\n",
+"//find i)Mean\n",
+"// ii)Variance of given function\n",
+"clc;\n",
+"clear;\n",
+"\n",
+"//Mean Value\n",
+"function X=f(x),\n",
+" z=3*(1-x)^2,//Marginal Probability Density Function\n",
+" X=x*z\n",
+"endfunction\n",
+"a=0;\n",
+"b=1;\n",
+"EX=intg(a,b,f);//Mean value of X\n",
+"\n",
+"function Y=c(y)\n",
+" z=3*(1-y)^2,//Marginal Probability Density Function\n",
+" Y=y*z\n",
+"endfunction\n",
+"\n",
+"EY=intg(a,b,c);//Mean value of Y\n",
+"disp(EX,'i)Mean of X =')\n",
+"disp(EY,' Mean of Y =')\n",
+"\n",
+"//Variance\n",
+"function X=g(x),\n",
+" z=3*(1-x)^2,//Marginal Probability Density Function\n",
+" X=x^2*z\n",
+"endfunction\n",
+"a=0;\n",
+"b=1;\n",
+"EX2=intg(a,b,g);\n",
+"\n",
+"function Y=h(y)\n",
+" z=3*(1-y)^2,//Marginal Probability Density Function\n",
+" Y=y^2*z\n",
+"endfunction\n",
+"\n",
+"EY2=intg(a,b,h);\n",
+"\n",
+"vX2=EX2-(EX)^2;//Variance of X\n",
+"vY2=EY2-(EY)^2;//Variance of Y\n",
+"\n",
+"disp(vX2,'ii)Variance of X');\n",
+"\n",
+"disp(vY2,' Variance of Y');\n",
+""
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.3: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Probability\n",
+"//Example 2.3\n",
+"//page no 43\n",
+"//find the probability \n",
+"clc;\n",
+"clear;\n",
+"total_cards=52;\n",
+"ways_of_drawingtwocards=52*51/(2*1);//ways of drawing 2cards from a deck of 52cards\n",
+"diamonds=13;\n",
+"Hearts=13;\n",
+"waysof_diamonds_Hearts=diamonds*Hearts;//ways of drawing a Diamond and a Heart\n",
+"probability=waysof_diamonds_Hearts/ways_of_drawingtwocards;\n",
+"\n",
+"disp(probability,'Probability that one card is Diamond and Othercard is Heart');"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.4: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Probability\n",
+"//Example 2.4\n",
+"//page no 44\n",
+"//find the probability \n",
+"clc;\n",
+"clear;\n",
+"whiteballs=4;\n",
+"blackballs=3;\n",
+"pa1=whiteballs/(whiteballs+blackballs);//probability of first ball is white\n",
+"pa2=(whiteballs-1)/(whiteballs+blackballs-1);//probability of second ball is white\n",
+"pa3=(blackballs)/(whiteballs+blackballs-2);//probability of third ball is black\n",
+"required_probability=pa1*pa2*pa3;\n",
+"disp(required_probability,'probability of first two balls are white and third ball is black');\n",
+""
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.5: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"\n",
+"//Caption: Probability\n",
+"//Example 2.5\n",
+"//page no 44\n",
+"//find the probability \n",
+"clc;\n",
+"clear;\n",
+"whiteballs=5;\n",
+"redballs=3;\n",
+"balckballs=2;\n",
+"totalballs=whiteballs+redballs+balckballs;\n",
+"pw=whiteballs/totalballs;//probability of gettingfirst ball is white\n",
+"pr=redballs/(totalballs-1);//probability of getting second ball is red\n",
+"pb=balckballs/(totalballs-2);//probability of getting third ball is black\n",
+"\n",
+"possbilecases=6;\n",
+"probabalityof_eachcase=pw*pr*pb;\n",
+"totalprobability=6*probabalityof_eachcase;//required probability\n",
+"disp(totalprobability,'Probability of balls will be in different colours');"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.6: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Probability\n",
+"//Example 2.6\n",
+"//page no 45\n",
+"//find the probability \n",
+"clc;\n",
+"clear;\n",
+"//Probability of solving the problem\n",
+"PA=3/4;\n",
+"PB=2/3;\n",
+"PC=1/4;\n",
+"//Probability of not not able solving the problem\n",
+"PNA=1-PA;\n",
+"PNB=1-PB;\n",
+"PNC=1-PC;\n",
+"//probability that no one will solve the problem\n",
+"probability_notsolve=PNA*PNB*PNC;\n",
+"//probability that problem willbe solved\n",
+"probability_solve=1-probability_notsolve;\n",
+"disp(probability_solve,'probability that problem willbe solved');"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.7: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Probability\n",
+"//Example 2.7\n",
+"//page no 45\n",
+"//find the probability \n",
+"clc;\n",
+"clear;\n",
+"whiteballs=3;\n",
+"redballs=2;\n",
+"totalballs=whiteballs+redballs;\n",
+"pw1=whiteballs/totalballs;//probability of first ball is white\n",
+"pr2=redballs/(totalballs-1)//conditional probability event second ball is red with first white\n",
+"probability=pw1*pr2//probability of second ball is red with first ball is white\n",
+"disp(probability,'probability of first ball is white and second ball is red');"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.8: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Probability\n",
+"//Example 2.8\n",
+"//page no 45\n",
+"//find the probability that recorded number is 2\n",
+"clc;\n",
+"clear;\n",
+"head=1/2;//probability of getting head\n",
+"tail=1-head;//probability of getting tail\n",
+"head2= head*1/6;//probability of getting head with recorded number is 2\n",
+"tail2=tail*1/36;//probability of getting tail with recorded number is 2\n",
+"\n",
+"//probability of getting recorded number is 2\n",
+"p=head2+tail2;\n",
+"disp(p,'probability of getting recorded number is 2');"
+ ]
+ }
+,
+{
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Example 2.9: Probability.sce"
+ ]
+ },
+ {
+"cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+"source": [
+"//Caption: Probability\n",
+"//Example 2.9\n",
+"//page no 46\n",
+"//find the probability \n",
+"clc;\n",
+"clear;\n",
+"PA=1/8;\n",
+"PB=1/12;\n",
+"probability_makingerror=1/10001;\n",
+"probability=(PA*PB)/((PA*PB)+((1-PA)*(1-PB)*probability_makingerror));\n",
+"disp(probability,'Probability of program is correct');"
+ ]
+ }
+],
+"metadata": {
+ "kernelspec": {
+ "display_name": "Scilab",
+ "language": "scilab",
+ "name": "scilab"
+ },
+ "language_info": {
+ "file_extension": ".sce",
+ "help_links": [
+ {
+ "text": "MetaKernel Magics",
+ "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
+ }
+ ],
+ "mimetype": "text/x-octave",
+ "name": "scilab",
+ "version": "0.7.1"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}
generated by cgit (git 2.34.1) at 2025-07-06 13:43:33 +0000