Lecture 20 : Random Variables: Continuous and Discrete

[Music] Hello friends welcome to the course multivariate procedure with r now from this lecture and in the next couple of lecture we are going to talk about the random variables and probability distributions and uh we are going to begin with a univariate case and then gradually I will extend it to a multivariate case now the next question comes why do we need a random variable the basic uh job of a statistics is to analyze the data now how the data is collected on what it depends on the objective suppose the objective is to know the age of the persons staying in a colony then what are you going to do you are going to collect the observations on age of those people who are staying in that Colony you are not going to collect the observation on the weight of those people who are staying in the same Colony also you are not going to collect the age of those people who are staying in some other area some other colony so if you try to see in this very simple example when the objective is to find out the average age of the people who are staying in a colony you are going to collect the observations on the age of people who are staying in that Colony so now what is age age is a variable on which you you are going to collect the observation similarly if you want to know the effect of a medicine in controlling the direct blood pressure what do you like to do you will try to give the dose of the medicine to a group of patients and then you will try to record the data on blood pressure you are not going to collect the data on the body temperature of those patients but in case if your objective is that how the medicine is going going to control the body temperature or fever of the people then what are you going to do then you are going to collect the data on the body temperature so you see based on the objective of the study you always try to Define a variable on which you are going to collect the data now when you are trying to collect the data on certain variable there is always going to be some random variation which is beyond our control suppose if I ask you that what is the content of nitrogen in the soil in a field do you agree with me that the content of the nitrogen of the soil in the entire field is going to be the same or similarly if I say what is the level of moisture in the soil in that field do you think that the level of moisture at every point in the field is going to be the same no there will be a small variation so that is why there is some uncertainty involved in the observation and that is why in practice or practically we say that we have got a random variable on which we are going to collect the observation now different questions crop off how should you define the random variable how should you measure this un uncertainty and uh and when the observations are going to have uncertainty their behavior will depend on the type of process what they have so how to characterize the behavior of the process through a mathematical function why do you need a mathematical function we understand and we believe that any result which is based on the foundations of mathematics is acceptable and it is well accepted at every place it is just like if I declare in my class that all the students who are wearing a blue shirt they will be getting a grade A and all the students who are wearing a say this white shirt they will be getting a Grade B and all those student who are getting say this grade c those who want to have grade C they have to wear suppose a black color shirt and so on do you think that this type of decision is going to be acceptable certainly not but if I say okay I will take your exam and I will give you some questions for which I have taught you in the whole semester and if you get more than 90% marks I will give you grade A in case if you say get a between 75 and 90% Mark then I will give you Grade B and so on do you think that this decision is going to be acceptable answer is yes now what is the difference between the two ways of of making decision in the second case there is an involvement of mathematics so that is why whenever we are trying to find anything from the data that has to be in some mathematical format and that is why whatever we are observing that has to be a numerical value whatever is the uncertainty involved that has to be defined in s in such a way which is mathematically compatible so now all these different aspects we are going to discuss in this lecture and in a couple of lectures in the future now in this lecture we are going to talk about the random variables so basically I can tell you we have a a very well defined definition of random variables which is based on strong foundations of mathematics but definitely uh the objective in the course is not to go into the details of mathematical statistics we have a topic major theory in which we try to define the random variables in a proper way using the proper mathematical definition but definitely here I'm going to make it in a very simple way so that you can understand it from the application point of view now in case if I try to broadly classify you can have have two types of observations for example if you roll a dice then there can be possibilities of number like 1 2 3 4 5 and six but if I ask you to measure the heights of the students that can be 155 CM that can be 155.0 CM that can be 150 51 1 cm and so on so you can see that in the case of height the values can be taken in a defined interval whereas in the rolling of a dice the values are going to be taken are only the integer values and beside this thing when you try to roll a DI then there is always the probability of occurrence of the event and in case if you try to Define your event then the probability also probability is defined and if the event changes then the probability of event also changes so so even what you are going to observe is something but you want to translate it to a mathematical function because the basic requirement in statistics is that we need the mathematical data or we need the values in mathematics on which we can do different types of mathematical operations so now we try to understand these things one by one and in this lecture let us try to understand the basic concept of continuous and discrete random variables so let us begin our lecture so now the first question comes here why do you need a random variable and this random variable can be univariate by variate or say multivariate univariate mean there is only one variable by variate mean there are two variables and multivariate mean there are say this more than one variable by variate is also a particular case of multivariate so so we know that some concepts are required to draw the statistical conclusion from a sample of data about a population of Interest why if you try to see what is the basic objective of Statistics statistics is always trying to observe a small sample of data and based on that it always try to take the some statistical conclusions which are valid for the entire population right for example in case if you want to know that uh what is the initial salary of students uh who are graduating in the St six subject from a university now what you can do that you can draw there a sample suppose I try to draw a sample of say 100 student who are going to complete their degrees in the statistics program and based on that we will try to find out that what is the average salary of these hand students but we are going to use it to define or to give the information for the whole population that mean the expected salary of all the students who are graduating or completing their degree with the statistic subject similar is the case when we are trying to deal with a clinical trial uh medicine is exposed or given to a group of people but whatever are the conclusions they are going to be valid over the entire population right Suppose there is some new medicine new drug that is given to a sample of selected patients and suppose some patients show Improvement and some patient does not show the Improvement so and our objective is that we want to understand what is the what are the different effects or Consequences of this medicine when it is given to the people in this country or in this world so that mean the population of patients so now in order to understand this thing in order to implement this concept that how to implement the conclusion based on a small sample to a whole population we need some basic statistical concept and among them the random variables are going to lay the foundation of the theoretical Concepts which are required to make such a statistical inferences and conclusion you would like to go for sampling you would like to go for estimation we would like to go for test of hypothesis which are the different components of the statistical inference and you will see the beginning of all these procedure is the random variable now let me try to give you an idea or an example through which I can explain you that what is an experiment what are the outcomes and how a random variable is associated and how it is transformed to a statistical function so let me take here a very simple example of coin where we toss the coin and whenever we toss a coin the outcomes are either either head or tail right so we know that the sample space in such a case which we are trying to denote by Capital Omega that's a a standard symbol for the sample space it is a collection of all such points which I'm trying to indicate by small Omega where Omega where this small Omega can be either head or tail right so I can say now here that okay the outcome of a crossing of a coin is either head or tail so just by writing St or head or tail do you think that is it really going to help us it is very difficult to handle the elements in statistics if they are not in numbers so now we need to do something so that we can convert this outcome of head and tail into some numbers which is in turn going to help us in doing different types of statistical calculations so now we suppose we Define this capital x to be a function such that when Omega is observed as head that is small Omega then the value of the X which is indicated by here x Omega takes one and if the outcome is tail then X Omega takes value zero so now if you try to see on one side you have here head and tail and on other side you have here the numericals some numbers and in statistics as soon as we have numbers we are in a position to implement our mathematical functions so you can see here that now X is the real valued function which is defined on the sample space Capital Omega but what is happening that Capital Omega has had and tail now it is bringing us to some other space where these are going to be indicated by here two numbers like this one for head and zero for tail so this type of transformation is needed and this type of transformation is being done by this function here x right so this x is called as random variable why this is called as a random variable because when you cross a coin then unless and until the experiment is completed you don't know whether you are going to get head or tail so there is some Randomness involved so now this head is denoted by one and tail by zero so this sample space Omega has now 2 points Zer and one right so you can see here earlier this Omega was here s and T but now Omega is transformed to a new set of values where which takes value 0 and one so now in any random experiment we are always interested in the value of some numerical quantity which is determined by the result we are not really interested in all the details of the experiment I'm not really interested that how you have tossed the coin whether 5T high or 7 ft high or in the 20° centigrade or say 100° Centigrades this is not my concern what I want I just want please give me the outcome head tail tail head Etc and these quantities of Interest they are determined by the result of the experiment and these are associated with this random variables now the because this these values are determined by the outcome of an experiment and we are trying to associate here a random variable whose outcome is not known with 100% confidence that if you toss a coin unless and until it comes on the ground you don't know whether it is going to be head or tail both the possibilities are there so suppose we assume or we say that okay there are 50% chances a head will come and and 50% chances the tail will come so in that case we can associate the probability with the outcomes of the random variable so you can see here that X is taking values here one and zero one forhead so if I say that x equal to 1 that means head is observed and the probability of observing xal to 1 that is the probability of observing here head this is 1 upon 2 0.5 and similarly the probability of observing tail that is the random variable X takes value zero is also here 1 by two so now we can view X as a random variable which collects the possible outcomes of the random experiment and capture the uncertainty associated with them right so this is how we are going to introduce the random variable so now if I try to translate whatever I have explained you so now in short I can say here that let Capital Omega represent the sample space of the random experiment that what are the possible values which can be observed and let this capital r be the set of real numbers right now I can Define or I can see that a random variable is a function X which assigns to each element small Omega belonging to Capital Omega one and only one number for example X X Omega is equal to small X small X belongs to r that is X is trying to map Capital Omega to R so that you can see here Capital Omega in the earlier example had two points H and T and R had here two points 1 and zero H is being transported to one and T being transported to zero so H and T are mapped to one and zero respectively right so before I go further with the details of uh this random variable and other aspect you C me try to explain you here what is the convention in statistic that how do we indicate the random variable and its values you know that when you are trying to define a random variable then you are going to get the observation on them for example suppose if I say that I want to have the observation on the heights of the students Heights of of a students so these Heights are going to be measured for different student and say suppose CM so I can Define the heights of the student as a random variable capital x and suppose now I now I obtain the height of say student number say here one and suppose it comes out to be suppose here 100 cm so this is indicated by here small X and this is the first observation so I can say here this is a small X1 and similarly if if I say that height of the St number two is suppose 180 cm then this can be denoted by here small X2 so now what I'm trying to do I'm trying to indicate the height of the student as capital X in capital alphabets and the associate values in lower case alphabets and this numbers are trying to indicate the observation number this the first observation second observation and so on similarly in case if I try to suppose I decide that okay let me try to also collect the weights of the students so now I can use here another variable here say here capital Y to indicate the weights of the student and suppose we are going to measure them in kilogram suppose the weight of the student number one comes out to be suppose here 50 kg so this will be denoted by here a small y1 and suppose the weight of the second stent comes out to be 60 kg then it is going to be indicated by small Y2 so if you try to see what I have done I'm trying to indicate the random variables by the uppercase alphabets and their values by the lower case alphabets so actually this is a convention in statistics to indicate or denote the random values by capital letter alphabets uppercase alphabets and their values by small letters say this lowercase alphabets so if I try to take the random variable to be here capital X then their values will be small X right so that is and if I try to write down here say suppose if I write down here let X1 X2 say x00 be the heights of students what does this mean that means we have got 100 numerical values or we have got the heights of the hand students which are some number which are some numerical values right okay now when we are talking about the random variables then we have two types of random variable one is called as discrete random variable RV means random variable and other is continuous random variable what are these things so let me try to uh first give you a brief idea and uh then I will try to give some more examples of these type of random variables so when you are trying to consider a random variable which is taking the values only at some points then it is called in simple word as discrete for example if I say the number of heads obtained when you are crossing a coin 100 times so you cross a coin 100 times try to count the number of HS once again try to Coss a coin 100 times count the number of heads so this number is always going to be an integer and it can take all possible values between 1 2 3 4 up to 100 but if I say there are 25.5 heads it's not there so all the values they are concentrated only at some points so the random variable whose set of possible values can be written as a finite sequin x1x2 xn or as an infinite sequin X1 X2 dot dot dot they are said to be discrete so the sample space of uh such a random variable that is said to be discrete if it consists of a finite or countable set of outcomes right for example in the example which I just explained you that you try to c a coin for 100 times and then try to count the number of heads in such a case the random variable it is a random variable whose set of possible values is the set of non- negative integers 1 to 100 and this is a discrete random variable so now we consider an example Suppose there is a c customer care phone and which has 30 external lines that means there are something like 30 telephone connection and different customers are calling to the customer care so at a particular time that some of the lines may be used and others may be free so at a given point of time suppose five customers are calling then five of the lines will get occupied if there are 10 customers which are calling then the 10 lines will be occupied and suppose if more than 30 customers are call calling at the same time then the all the 30 lines will be used and the customer will be asked to wait right so now at a given point of Time how many telephone lines are being used that is not known to us so this can be indicated by a random variable X so now can see here x can take difference between different values uh say from 1 to 30 that means only one line is being used or two lines are used or 30 lines are used now suppose if I say that at a particular time there are five telephone lines which are being used that means the value of the capital x here is five so this is going to be indicated by small X so I can say here if five lines are used then small X will be equal to five and this is how we are going to interpret it okay now I have given you a faar idea about the discrete random variable now I come to continuous random variable suppose the length or temperature fluctuation or calibration cutting tool Veer bearing weer Etc they have to be measured right and they are going to be measured in some units for example the temperature fluctuation that can be in degree Centigrades or degree Fahrenheit whereas those cutting tool we are mean that means when you when you are trying to use an equipment it gets distorted so how much it has been distorted after being used that can be measured in some units like how that how much material is reduced in grams or volume Etc but anyway you can see that whenever you are trying to measure such observations then what is going to happen first of all there will be small variations in the measurements and this variations may be due to different reasons for example uh just now I had given you an example that if you want to measure the moisture level in the soil in a field then the level of moisture contain in different parts of the field it is not going to be same but there will be small variation so now in such a situation if you try to represent by capital x the quantity which you are going to measure then you cannot say that the random variable is going to take the values only at some finite values for example as in the case of discrete we say that if there are five telephone lines are going to be used then the value of the random variable is five and if a eight lines are going to be used then the value of the random variable is eight but in the case of this moisture level in the soil or say temperature this can be 30° CG 30.1 de Cen 30.1 1° C 30. 111° Centigrades and so on so there will be some random variation and these values are going to be defined as if the random variable is going to assume the values in in the form of an interval of real numbers right so if I try to consider that for any equipment you are going to measure the length temperature fluctuation calibration etc etc this type of model will give us an idea about the Precision in the length measurements but these values are going to be measured on some continuous scale there will always exist values between two values that is what I mean to say so now I can say that uh there exist random variable that take on a Continuum of possible values and such variables are known as continuous random variables and another example I can give you is the for uh measuring the lifetime of a bulb that can be one day one day one hour one day 1 hour one second Etc so in case if you want to make an assumption on the bulb's lifetime then we can assume that the lifetime of the bulb is lying in some interval say A and B where A and B both are positive right so in such a case we have a concept of say this continuous random variable and in both the we have considered the univariate case that mean there are only there is only one variable now we are going to consider a situation where the process is going to be affected by two random variables so we call them as a by variate random variable so these two random variables they can be independent they may be dependent what does this mean the two random variables are going to be independent if the occurrence or non-occurrence of one variable does not affect the occurrence or say nonoccurrence of the second variable and they are dependent that if the occurrence in the first variable affect the outcome in the second variable right well we will understand that how to define such a Independence through the random variable in the forcoming lectures and both these random variables they can be continuous means both are continuous both are discrete or a combination of them that one of them is continuous and another is discrete so this is the general setup of a by variate random variable right for example in case if I say I want to measure the height and weight of a person so this can be indicated by saying that let X1 indicates the height and X2 indicate the weight of a person and both we know that height and weight they are the positive value so I can say that the person's height and weights are assumed to take on any value in the interval A1 B1 and A2 B2 respectively right so height lies between A1 B1 and weight lies between A2 B2 for example height can be between suppose say 50 cm to suppose 1770 cm and the weight can be suppose here 10 kg to here 60 kilog right so these are some numerical values so now the question is this in statistics how do we we express such by variate random variable so it is indicated by a 2 cross one vector now this is a mathematical vector vector like vectors and matrices it is not a data Vector of the r software so we try to indicate here as say here it's a vector of random variable X1 and X2 and we write this here as a xcore is a 2 cross one vector or I can write down here as a X1 X2 transpose and when we are trying to have this two random variables then some other properties also come into existence now if you try to see I I started with a unived random variable I extended to two random variable and similarly I can extend to a general setup where we have more than two random variables right and this setup will be called as multivariate random variables so Suppose there are more than two random variables and once again all the random variables they can be independent they can be dependent and we have a statistical measur by which we can actually judge and it is possible that all the random variables they can be continuous they can be discrete or they can be a combination that some are continuous and some some are discrete so this is the setup of a multivariate random variable right so say one simple example can be if I try to extend the example of height and weight that suppose I want to measure measure the this different parameters of a human being so suppose I want to measure the person's height weight age blood sugar and blood pressure so there are going to be height can be measured by X1 weight can be measured by X2 a can be measured by X3 blood sugar can be measured by X4 and blood pressure can be measured by X5 right and suppose these all these variables will have some numerical values which are going to be assume that let that height which is indicated by X1 it is lying between A1 B1 second variable X2 is lying between A2 B2 third variable is lying between A3 B3 fourth variable is lying between A4 B4 and fifth variable is lying between A5 B5 and for example I can assume depending on the situation suppose I assume that all Ai and V they are greater than zero right although I am taking here in example where all the variables I'm assuming to be continuous but they can be discrete also or it is possible that suppose X1 X2 X3 are continuous and X4 and X5 are discrete that is also possible but here my objective is to give you the idea of uh what is a multivariate random variable so now all these five variables they can be expressed in a vector which is of order 5 by one and the way I had expressed X1 and X2 similarly I can extend it to X1 X2 X3 X4 X5 or this is a transpose of X1 X2 X3 X4 X5 now in case if you want to uh take the observations so suppose if I consider here the by cutup so suppose if I say here person number here one and say person number here two and suppose the height of the person number one comes out to be 150 cm and the weight of the person number one this comes out to be suppose here 50 kg and for the person number two suppose the height comes out to be 160 cm and the weight is suppose 60 kg so now these are the observations which are going to be indicated by X1 underscore like this and the second observation is going to be like here X2 uncore right similarly if you try to consider here a multivariate setup so now you will have suppose I have the data of here person number one and person number two for the person number one you will have here a vector of 5 by1 height weight age blood sugar blood p PR and similarly for the person number two height weight age blood sugar and blood pressure and this will be indicated by here xcore one and uh the data of the second person will be indicated by xcore two so that is indicating that these are the observations on the here five variables right so now I can now Express in general that suppose we have got here P random variables which are indicated by Capital X1 Capital X2 up to here Capital XP so all these variables can be defined in a in the setup of a p cross one random vector and this Vector can be indicated by here as xcore and which will be here A P cross one vector consisting of values X1 X2 XP and which can also be written as say here X1 X2 XP transpose now in case if you want to get here the first value then it will be like as a first value on X1 first value on x2 first value on here see here XP like I can say X11 X12 up to here X1 p and similarly you can have here the second value third value and so on so in such a case the space of X this random Vector will consist of of set of n tuples that means n sets of observation if I see here we are getting here X1 X2 xn like this okay so now we come to an end to this lecture and uh you can see here uh that was a pretty Elementary lecture but my objective was to expose you with the concept of random variables because you will see that in all the statical tool our first sentence is going to be let X be a random variable that can be a unived random variable that can be a bate random variable or that can be a multiv random variable it depends on the requirement and objective of the study that what we really want to do and depending on the setup that depending on the choice of random variables our statistical tools are going are also going to differ in case if my random variable is continuous then the way I am going to operate or find out different types of tools for example mean or even variance that is going to be a different way in comparison to when my random variable is discrete and suppose if we have a combination then we have to see that how are we going to compute different statistical quantities now depending on the nature of random variable we have different types of characterization of the probability function and based on that all other tools they also change so now but at this moment my request to you will be try to have a loop in some books and try to read more about this random variables once again again I will say the way I have explained you the random variable Concept in this lecture this is a very Elementary one but my objective was to expose you so that I can take you forward to a multivarious setup in case if my objective is to give you a course on probability Theory or major Theory possibly I will give you all those basic definitions here and believe me we have very concrete solid definitions which are based on Hardcore mathematics for uh random variables and different types of concept that we are going to use further but once again I will say that I'm not going into the mathematical details well I have limited time here but I will request you that you try to now start reading a book on any this probability Theory or the probability distribution mathematical statistics whatever book you want uh there are books which are varying from lower level of mathematics to a hardcore mathematics and this is the how this is the way uh by which we both can go together and the learning and and in the same time the learning process can make you learn better you will understand More Concept so you try to read from the book try to understand these concept and I will see you in the next lecture till then [Music] goodbye [Music] [Music] [Music]