 Hello and welcome to the session. I am Deepika and I am going to help you to solve this question. The question says let X denote the sum of the numbers obtained when two fair ties are rolled, find the variance and standard deviation of X. Now we know that the mean of the variable X is given by E X is equal to mu which is equal to summation of X i P i where i varying from 1 to n and variance of X is given by E X square minus E X whole square and standard deviation of X is given by sigma X is equal to under root of variance of X. So this is a key idea behind our question. We will take the help of this key idea to solve the above question. So let's start the solution. Now in this question we are given let X denote the sum of the numbers obtained when two fair ties are rolled. So the sample space of the given experiment has 36 assignments. Now here in each ordered pair the first digit shows the number when first tie is rolled and the second digit shows the number when the second tie is rolled. So we can write the sample space of the given experiment has 36 assignments. Now we are given let X denotes the sum of numbers obtained when two fair ties are rolled. So the random variable that is the sum of the numbers two ties takes the values 4, 5, 6, 7, 8, 9, 10, 11 or 12. Now we have to find the variance and standard deviation of X. So for this we will first find out the probability distribution of X. So the probability distribution then the sum of the numbers is 2 when two fair ties are rolled then 3X is equal to 1 over 36 because out of 36 ordered pairs only one ordered pair has sum of the numbers as 2. Similarly for X is equal to 3 we have PX is equal to 2 over 36 because out of 36 ordered pairs only two ordered pairs has the sum of its numbers as 3. Again for X is equal to 4 PX is equal to 3 over 36. Again for X is equal to 5 PX is equal to 4 over 36. Again for X is equal to 6 PX is equal to 5 over 36. X is equal to 7 PX is equal to 6 over 36. Again for X is equal to 8 PX is equal to 5 over 36. Similarly for X is equal to 9 PX is equal to 4 over 36. Now for X is equal to 10 PX is equal to 3 over 36 and for X is equal to 11 PX is equal to 2 over 36 and for X is equal to 12 PX is equal to 1 over 36. So this is the probability distribution of X. Now, for variance of x, first we will find out the expectation of x that is the mean of x. Now, according to our key idea, we know that the mean of variable x which is denoted by mu is equal to sigma i varying from 1 to n. Now, we know that the mean of a random variable x is also called the expectation of x. So, we have E x is equal to mu which is equal to 2 into 1 over 36 plus 3 into 2 over 36 plus 4 into 3 over 36 plus 5 into 4 over 36 plus 6 into 5 over 36 plus 7 into 6 over 36 plus 8 into 5 over 36 plus 9 into 4 over 36 plus 10 into 3 over 36 plus 11 into 2 over 36 plus 12 into 1 over 36 and this is equal to 2 plus 6 plus 12 plus 20 plus 30 plus 42 plus 40 plus 36 plus 30 plus 22 plus 12 over 36 now this is equal to 252 over 36 which is again equal to 7. Now, E x square is equal to 2 square into 1 over 36 plus 3 square into 2 over 36 plus 4 square into 3 over 36 plus 5 square into 4 over 36 plus 6 square into 5 over 36 plus 7 square into 6 over 36 plus 8 square into 5 over 36 plus 9 square into 4 over 36 plus 10 square into 3 over 36 plus 11 square into 2 over 36 plus 12 square into 1 over 36 now this is again equal to 1 over 36 into 2 square into 1 that is 4 into 1 plus 3 square into 2 which is 9 into 2 plus 16 into 3 plus 25 into 4 plus 36 into 5 plus 49 into 6 plus 64 into 5 plus 81 into 4 plus 100 into 3 plus 121 into 2 plus 144 into 1 and this is again equal to 1 over 36 into 4 plus 18 plus 48 plus 100 plus 118 plus 294 plus 320 plus 324 plus 300 plus 242 plus 144 and this is again equal to 1,974 over 36 which is again equal to 54.833 now according to our key idea variance of x is equal to E x square minus E x whole square so this is equal to 54.833 minus 7 square so this is equal to 54.833 minus 49 which is again equal to 5.833 so the variance of x is equal to 5.833 now again the standard deviation of x that is sigma x is equal to under root of variance of x and this is equal to under root of 5.833 and this is equal to 2.415 now the answer for the above question is variance of x is equal to 5.833 and standard deviation of x is equal to 2.415 this completes our session I hope the solution is clear to you why and have a nice day