Problem 4.3 (Video 3.1, 3.2, 3.3, 3.4, Quick Calculations) Calculate each of the requested quantities. All of these problems are carefully chosen so that they can be completed without integration, so we are expecting exact answers. For the Gaussian problems, you will sometimes need to lookup values for the standard normal CDF (z). For example, you could use a lookup table https:/ /en. wikipedia.org/wiki/ Standard_normal_table or the MATLAB function normcdf. (a) For fx (x) = 1 - 2 12 51 0 sketch the PDF, then calculate E[X] and P[|X| > 1/2]. otherwise (b) For fx (x) from part (a), determine and sketch the conditional PDF fx B(x) given the event {IX| > 1/2}. Using your sketch, calculate P [X > 0 [X| > 1/2]. (c) Let X be Exponential(2). Calculate E[X - 1] and E[(X - 1) 2]. (d) Let X be Uniform(-2, 3). Determine P[X - 1 > 0] and P[X2 - 1 > 0]. (e) Let X be Gaussian (2, 1) and let Y = 3X - 2. Determine the mean and variance of Y as well as P[Y > 1]. (f) Let X be Gaussian (1, 4). Determine (up to two decimal places) the maximum value of a such that P[X b]