Variance o2 = E[(X - u)?] (where u = EX ) tells you how far the distribution is spread out of from its mean. Similarly, E [(X - u)4/o'] which is called Kurtosis, tells you how heavy the tail of your distribution is. We say a distribution has a heavy tail if its density is relatively large on the points that is far away from the mean. For example, kurtosis of exponential random variable and normal random variable are 6 and 3, respectively. In fact, exponential distribution has heavier tail than normal random variables as you can see in the following figure. 0.5 Standardized Exp(1) Standardized Exp(1) N(0, 1) N(0, 1) 0.4 density 0.000 0.005 0.010 0.015 0.020 0.025 0.030 density 0.2 0.1 0.0 3.0 3.5 4.0 4.5 5.0 In this question, you will show that kurtosis of normal random variables is 3, using some facts you learned in this class. You may find page 358 - 360 of the textbook helpful. Suppose X, Y " N ( 1, 02). You may use the result from previous parts even if you couldn't solved them. (al) (1pt) What is the density of R = V(X-4) + (Y-4) and what is the name of this distri- bution? Proof is not required. (a2) (1pt) What is the density of S = R2 ? Provide the name and the parameter of this distribution. Proof is not required. (b) (4pts) Show that ES2 = 8. (c) (4pts) Show that E (-H)* = 3. (Hint : Expand S2 and use the fact that X, Y are independent)