Task 1 PDF [15 ptsl For this exercise. give exact answers as simplified fractions. Define function fx by ._._ (143\"? ifl-{m-cl fall-l _ l 0 otherwise a) Show that Ix has the properties required of a probability density function. [1) Compute the expectation of a random variable X with y; as its PDF. c) Compute the variance ofa random variable X with Ix as its PDF. Task 2 How long? |8 ptsI Suppose that the distance from Seattle to Portland is 180 miles. You decide to buy an electric bicycle for the trip. Suppose that electric bikes have speeds that are uniformly distributed between 15 and 30 miles per hour, and you buy a random motorized bike {i.e.. its speed is Uniform(15.30)). Let T be the time it takes you to ride from Seattle to Portland. What is EFF]? Note: Recall that the indefinite integral of g(.r.] = 1-" is equal to 111p.) + 0'. assuming J.' r: ll. Task 3 Practice with uniforms [12 pts] Suppose that X w Uniflll. 1). Let aJ} be constants such that (l b|X > a)? b) (3 point) For any real number 3;, what is 1P{X c le .'> a)? Be sure you include all cases. C) (6 points) Let Y be a random variable such that lm = lPlX c le 2 a} What is fy-{yll and what kind of random variable is Y? {Note that when you specify a distribution from our zoo. you must specify both the name ofthe distribution and the values of the parameters of that distribution.)