 Hello and welcome to the session. In this session, we will define a random variable for a quantity of interest by assigning a numerical value to each event in our sample space. Then, we will graph the corresponding probability distribution using the same graphical display as for data distribution. First of all, we shall define a random variable. It is a variable whose value is determined by the outcome of a random experiment. The letter capital X is used to represent the name of random variable and the letter small x is used to represent the value of random variable. The probability that the random variable capital X will be equal to small x is given by probability of capital X is equal to small x or probability of small x. Now let us discuss some examples. Here we take the first example. Suppose we toss a coin two times then sample space will be given by the set containing h h h t t h t t where h denotes the occurrence of head and t denotes the occurrence of tail so here h h denotes the occurrence of two heads h t denotes the occurrence of a head then the occurrence of a tail t h denotes the occurrence of a tail followed by occurrence of a head t t denotes the occurrence of two tails let capital X represents the number of tails that result from this experiment then values for x will be 0 1 and 2 that is we assign numerical value to each sample point and for h h we assign 0 for h t we assign 1 for t h we assign 1 and for t t we assign 2 here capital X is the random variable because its value is determined by the outcome of an experiment. Now let us draw probability distribution table for the random variable capital X now this is the probability distribution table here we see that probability of capital X is equal to 0 is equal to 1 by 4 as we have one outcome favorable to the event of no tail occurring and total number of outcomes in this event is 4 so we have the probability 1 upon 4 similarly we have the probability of capital X is equal to 1 as 2 upon 4 and the probability of capital X is equal to 2 as 1 upon 4 now let us draw its graph this is the required probability distribution graph where we have taken small x values along horizontal axis and probability values along vertical axis also when small x is equal to 0 we see that the probability is equal to 0.25 but small x is equal to 1 the probability is given by 0.5 and when small x is equal to 2 the probability is again equal to 0.25 let us consider another example which says two balls are drawn at random in succession without replacement from a bag containing three red balls and five blue balls then the possible outcomes can be the set containing r r r v v r and b v here r r denotes that the two balls drawn are red in color and they are drawn one after another r v denotes that the first ball drawn is red in color followed by a blue ball we are denotes that the first ball drawn is blue in color followed by a red ball and b v denotes the two balls drawn are blue in color and they are drawn one after another now let capital X denotes the number of red balls in the outcome let us draw a table for all possible outcomes and the numerical values assigned to them so here we have assigned two for r r one for r b one for b r and zero for r r now we shall find the probability of getting two red balls when we draw out the balls one at a time so here probability of first ball being red will be equal to number of favorable outcomes that is here we have three red balls so number of favorable outcomes will be three upon total number of outcomes that is equal to eight so we have the probability of first ball being red as three upon eight now probability of second ball being red will be equal to two by seven because there are two red balls left in the bag out of a total of seven balls left so probability of two red balls when we draw out the balls one at a time will be equal to three by eight into two by seven and this is equal to two ones are two two fours are eight so we get three upon twenty eight which is approximately equal to zero point one one so here we get the probability when capital X is equal to small x that is two will be equal to zero point one one one now we shall find the probability of drawing a red ball for this case we will add up the probabilities that is we will add up the probability of first ball being red followed by a blue ball and the probability of first ball being blue followed by a red ball so we have probability of first ball being red followed by a blue ball is given by now probability of first ball being red will be three by eight into probability of second ball of blue color will be equal to five upon seven because there are five blue balls still in the bag and seven balls all together after one red ball is being drawn so probability of first ball being red followed by a blue ball will be equal to three upon eight into five upon seven that is equal to fifteen upon fifty six and this is approximately equal to zero point two seven similarly probability of first ball being blue followed by a red ball will be given by now probability of first ball being blue is five upon eight into probability of second ball being red after a blue ball is drawn will be equal to three upon seven so we have probability of first ball being blue followed by a red ball is equal to five upon eight into three upon seven that is equal to fifteen upon fifty six and this is approximately equal to zero point two seven so probability of one red ball being drawn will be equal to probability of r b that is first ball drawn is red followed by a blue ball plus probability of b r that is probability of first ball being drawn is of blue color followed by a red ball and this is approximately equal to zero point two seven plus zero point two seven which is approximately equal to zero point five four therefore the probability when capital X is equal to one is equal to zero point five four similarly probability of getting two blue balls when we draw out the balls one at a time and this is equal to five upon eight into four upon seven and this is equal to five upon fourteen which is approximately equal to zero point three eight so probability of capital X is equal to zero is equal to zero point three eight here we should note that the probability when capital X is equal to two plus probability of capital X is equal to one plus probability of capital X is equal to zero is equal to one that is probability of RR plus probability of RB plus probability of BR plus probability of BB is all equal to 1. Now let us see the graph for this probability distribution. Now this is the probability distribution graph in which we have taken the values for small x along horizontal axis and probability values along vertical axis. Here probability of capital X is equal to 0 is shown as 0.38. The probability of capital X is equal to 1 is shown as 0.54 and the probability of capital X is equal to 2 is shown as 0.11. Thus in this session we have defined a random variable for a quantity of interest by assigning a numerical value to each event in our sample space. Then we have drawn the graph for the corresponding probability distribution using the same graphical display as for data distribution. This completes our session. Hope you enjoyed this session.