 Hello and welcome to the session. In this session, we will calculate the expected value of a random variable and interpret it as the mean of the probability distribution. Now we shall discuss the meaning of expected value. The expected value is also known as population mean. The expected value of a random variable indicates its average or central value. The expected value of a random variable x is denoted by e of x or mu. If capital X is a random variable with possible values denoted by x i that is small x i and probability of small x i denotes probability of capital X is equal to small x i that is the probability distribution then expected value of capital X is given by e of capital X is equal to mu is equal to summation of small x i into probability of small x i where i goes from 1 to n e of capital X that is the expected value of random variable capital X is also called as mean of probability distribution. Let us consider an example. Find expected value of a fair die-throw in an experiment of rolling a die the possible outcomes are 1, 2, 3, 4, 5 or 6. So the values for x i that is small x i will be equal to 1, 2, 3, 4, 5, 6. Now we draw a table for small x i probability of small x i small x i into probability of small x i. So here in first column we write the values for small x i that is 1, 2, 3, 4, 5, 6. Now we shall write the values for probability of small x i. Now when small x i is equal to 1 that is after rolling a die we get a 1 then probability of getting a 1 will be 1 upon 6 as there are 6 total outcomes. Similarly probability of getting a 2 on rolling a die will be equal to 1 by 6 and this way we find all the other probabilities and we get these values for probability of small x i. Now we find the values for small x i into probability of small x i. For small x i is equal to 1 we have 1 into 1 by 6 that is equal to 1 by 6. Similarly we have written all the other values for this column. Now summation of small x i into probability of small x i will be equal to 1 by 6 plus 2 by 6 plus 3 by 6 plus 4 by 6 plus 5 by 6 plus 6 by 6. On adding all these values we get 21 by 6 that is equal to 3, 2, 6, 3, 7 for 21 so we get 7 by 2 which is equal to 3.5. So we can say that a fair die has an expected value of 3.5. Let us see one more example. The probability distribution table gives the probability of getting a specified number of heads if a coin is tossed 4 times. What is the expected number of heads if a coin is tossed 4 times? For finding the expected number of heads if a coin is tossed 4 times we will find summation of small x i into probability of small x i. For the given table let us find small x i into probability of small x i. Now for small x i is equal to 0 and probability of small x i is equal to 1 by 16. Here we have 0 into 1 by 16 that is equal to 0. Now the next value will be 1 into 4 by 16 which is equal to 4 by 16. Next we have 2 into 6 by 16 that is equal to 12 by 16. Now 3 into 4 by 16 which is equal to 12 by 16 and lastly we have 4 into 1 by 16 which is equal to 4 by 16. On calculating summation of small x i into probability of small x i we get 32 by 16 that is equal to 2. So expected value of capital X will be equal to 2. Hence the expected value of heads if a coin is tossed 4 times is 2. Thus in this session we have calculated the expected value of a random variable and we have interpreted it as the mean of the probability distribution. That's all for this session. Hope you have enjoyed this session.