1. Suppose X has the probability density function _ k(2x), forOSxSZ, fX (3:) _ { 0 otherwise. (a) Find k so that fX(a:) is a probability density function. (b) Derive the cumulative distribution function FX(:c). (0) Calculate P[0.5 S X S 1.5] by integration. ((1) Calculate P[0.5 S X S 1.5] by using the cumulative distribution function. (e) Calculate E[X] and Var[X]. 2. Let the random variable X be distributed as Binomial Bin(100,0.55). ax and 0;; are the mean and standard deviation of X respectively. You may use R for some of the calculations, but please also write out the relevant formulas in order to get the full marks for this question. (a) 03101113138 PDJ'X O'X S X S x + Ox]. (b) Calculate PM}; 0X 5 X S pix + 0X] using the normal approximation for the binomial. (c) Calculate P[,uX 0X 3 X g ,uX + 0X] using the Tchebyshev inequality. ((1) If X happens to be distributed as Binomial Bin(100,0.05) instead, would the normal approximation be appropriate or perhaps some other approximation