 Hello and welcome to the session, the given question says given that x bar is the mean and sigma square is the variance of n observations x1, x2 up to xn proves that the mean and variance of the observations ax1, ax2 up to axn are ax bar and a square sigma square respectively for a0 equal to 0. Let us now start with the solution the n observations are x1, x2, x3 and so on up to xn and the mean which is denoted by x bar of these observations is given by x1 plus x2 goes on up to plus xn upon the number of observations which are n in number and the variance which is denoted by sigma square is given by summation i running from 1 to n xi minus x bar whole square upon n. Let this be equation number 1 and this be 2. Now the new observations are ax1, ax2, ax3 and so on up to axn that is we have multiplied a to all these observations. So the new mean is equal to ax1 plus ax2 plus so on up to axn upon n taking a common from the numerator we have x1 plus x2 plus up to xn upon n and from 1 this is equal to x bar since the mean of n observations x1, x2 up to xn is given by x1 plus x2 plus xn upon n. Thus the new mean is ax bar. Now let us calculate the new variance this is equal to summation i running from 1 to n axi minus ax bar whole square upon n. Now taking a square common and outside the summation sign we have a square summation i running from 1 to n xi minus x bar whole square upon n and this from equation number 2 we know that as the variance of observations x1 to xn just denoted by sigma square hence the new variance when the observations are multiplied by a plus a square sigma square. So this completes the session hope you have understood it by and take care.