 Today in this session we want to study random variable. Learning outcome of this topic is at the end of this session student will be able to explain the concept of random variables. The contents of these topics are first we see what is the means of random variable then we study types of random variable there are two types one is a discrete random variable and second one is continuous random variable we want to see in details. Random variable is a variable that assumes numerical values associated with random outcomes of an experiment where one and only one numerical value is assigned to each sample point means here for the each sample point only one value we can assign and we get the output that is called as random variable. A random variable there is a property how we denoted that random variables there are some properties we can assume here. A random variable is denoted with a capital letter the probability distribution of a random variable X tells what the possible values of X are and how probabilities are assigned to those values. A random variable can be discrete or continuous types of random variables there are basically two types of random variables first discrete random variables second continuous random variable now we study in detail one by one that first we want to study here discrete random variable. Discrete random variable a discrete variable is a variable whose value is obtained by counting. For example for the discrete random variables are number of students present in the class number of red marbles in a jar number of heads when flipping the three coins students grade level if you observe these examples we get here one numerical value only one numerical value that is a property of random variable and the discrete random variable means what we get variable whose value is obtained by counting for example student present red marbles flipping the three coins students grade level we can obtain by counting that's why these are the examples of discrete random variable. Now problems on this discrete random variable a pair of fair dice is rolled let X denote the sum of the number of dots on the top faces then calculate construct the probability distribution of X second find probability X greater than equal to nine third find the probability that X takes an even value now we solve the first one that is construct the probability distribution of X now here we observe the possible outcomes of this event is 11 12 13 14 15 16 means what there are how many dice are there two dice two dice one dice shows one second dice shows there is a possibility of one second one one two third one one three one four in this way same first one now the second event it will show the two second one one in this way 21 22 23 24 25 26 means in this way we get 616263646566 in this way we get this possible outcomes the possible values for X are the numbers 2 through 12 X equal to 2 is the event 11 11 so p of 2 is equal to what 1 by 36 means what here total possibility total possibility is how much 36 therefore add the event to the event 2 means what 1 by 36 we apply here the basic definition of probability that is probability is equal to number of events divided by number of samples here how many number of samples 36 and how many number of events that's a 2 comes 1 plus 1 that is a 1 1 by 36 now for the same here we apply x equal to 3 means what x equal to 3 means what sum we want to get the sum x equal to 3 when we get 1 2 and 2 1 therefore number of events are there how much 2 and number of samples 36 therefore apply the basic definition of probability here probability of 3 is equal to 2 by 36 therefore the for the each event here in the table we calculated that is x event and respect you for that one probability apply the basic definitions and we got this one for the x equal to 2 we get probability of x equal to 1 by 36 for 3 it will 2 by 36 for 3 by 36 5 4 by 36 for the 6 5 by 36 for 7 6 by 36 for 8 5 by 36 for 9 4 by 36 for 10 3 by 36 for 11 2 by 36 and for 12 it is a 1 by 36 this is the solution for first now second what is the second question the event x greater than equal to 9 is the union of the mutual exclusive events of x equal to 9 x equal to 10 x equal to 11 and x equal to 12 why we are getting up to here 12 only because the maximum outcome of this first one and second one is 6 their sum is what 6 plus 6 is 12 now what we want to obtain probability x greater than equal to 9 then what is the probability for the solution is probability of 9 plus probability of 10 plus probability of 11 plus probability of 12 therefore it will become as 0.27 and it will be the property yes that is less than 1 now third one the probability that x takes an six different even values that's six different even values means what the out of this probability samples we can take an even values which one that one probability of 2 probability of 4 probability 6 probability of 8 probability of 10 and probability of 12 and we can write basic definitions and from that table probability of 2 is what 1 by 36 probability of 4 is 3 by 36 probability 6 is 5 by 36 probability of 8 is 5 by 36 and probability of 10 is 3 by 36 and probability of 12 is 1 by 36 therefore it will become as a 0.5 and it will obviously property yes it will be less than 1 now the diagram we can draw the diagram from this equation that is a probability distribution of x now probability this diagram is shows probability versus outcome now probability shows is the y-axis and outcome shows is the x-axis now we draw the graphs and this graph shows the probability distribution of a discrete random variable now here if you observe the probability is always less than 1 and starting from the 0 that is 0 0.05 0.10 0.15 0.20 and with respect to outcomes we can draw this diagram and this diagram is related to the probability distribution of a discrete random variable now continuous random variable a continuous variable is a variable whose value is obtained by measuring for example height of a student in a class weight of a student in a class time it takes to get to school distance traveled between the classes these are the continuous random variables example this is obtained by measuring and for the continuous random variable we can measure we can by counting now this equation for an continuous random variable with probability density function f of x this is a equation f of x into d of x equal to 1 that is all the x values range now we want to study the example of a continuous random variable what is that continuous random variable x is a continuous random variable with probability density function is given by function f of x equal to c of x and the range of for x is 0 less than x less than equal to 1 where c is constant find c now friends before going to the continuous random variable always remember the continuous random variable we calculate with the help of the integration here we take the each value for the discrete random variable we take only countable value but for the continuous random variable we can we can take the values from the range that's why here we use the integration now here recall the continuous distribution function formula now continuous distribution function formula is integration of f of x into d of x equal to 1 and that f of x is all the x range now put the values what is f of x f of x equal to c of x into d of x equal to 1 and what is the range of that integration it will lies between 0 to 1 and up see everyone observe that one and give that one range 0 to 1 and put the value of c of x into d of x equal to 1 then what we get the value of c now you know that right hand side c we want to calculate here value of a c you know the they are given the 0 to 1 that is a x value and right hand side we know that is a 1 and can you find out from that one value of the c yes we find out that from value of c what is that one c here put the values range what is the range range of f of x into d of x equal to 0 to 1 that way here apply here 0 to 1 and put the value of f of x equal to what c of x into d of x and what is equal to 1 therefore c of x integration integration after the doing the integration we get c by 2 is equal to 1 that is c is equal to 2 now references for this topics are thank you