It is well-known that ridge regression tends to give similar coefficient values to correlated variables, whereas lasso may give quite different coefficient values to correlated variables. We will now explore this property in a very simple setting. Assume n are the number of training samples, p the number of dimensions, x in the input and y the output. Suppose that n = 2, p = 2, x11 = x12, and x21 = x22. Furthermore, suppose that y1 + y2 = 0 and x11 + x21 = 0 and x12 + x22 = 0, so that the estimate for the intercept in a least squares, ridge regression, or lasso model is zero: βˆ 0 = 0. (a) Write out the ridge regression optimization problem in this setting. (b) Argue that in this setting, the ridge coefficient estimates satisfy βˆ 1 = βˆ 2. (c) Write out the lasso optimization problem in this setting. (d) Argue that in this setting, the lasso coefficients βˆ 1 and βˆ 2 are not unique – in other words, there are many possible solutions to the optimization problem. Describe these solutions

Complete the worksheet. use 3.14 for π and round all decimals to the nearest hundredth.find the volume of the composite three-dimensional figure above. important terms are written in bold.1. the pyramid has a square base that is inscribed in the circular base of the cone. draw an image of the composite shape of the base. 2. if the side length of the square base is 8 units long, what is the diameter of the circular base of the cone from the picture in number 1? 3. now that the diameter has been calculated, what is the circular area of the cone? ignore the square base of the pyramid.? 4. what is the formula for volume of a cone? 5.to calculate the volume of the cone, the height is still needed. if the cone has a lateral surface with a length of 13 units, what is the height of the cone? 6.now find the volume of the cone. do not include the empty space for the pyramid. 7.. the side length of the square base of the pyramid is 8 units long. what is the base area of the pyramid? 8.what is the formula for the volume of a pyramid9. since the cone and the pyramid share the same vertex, what is the volume of the pyramid10.using the volume of the cone and the pyramid, find the volume of the composite shape. remember, the pyramid is empty. 11,. referring to the picture in number 1, what is the area of the composite base knowing that the square is empty? 12what is the formula to find the lateral surface area of a cone? (13find the lateral surface area of the cone14what is the lateral surface area shape of the pyramid? what is the formula for its area? 15to find the lateral surface area of the pyramid the length is still needed. calculate the lateral surface length of the pyramid. 16 now that the lateral surface length is known, what is the lateral surface area of the pyramid17. using the lateral surface areas of the cone and pyramid, along with the area of the composite base, what is the total surface area?