Consider a continuous random variable X with the following PDF: {: -1 ≤ x ≤0 or 1≤ x ≤ 3 otherwise (a) Sketch the PDF of X. Be sure to label your axes. (b) Determine the value of c that satisfies the normalization property. Set c to this value for the remainder of the problem. This can be done without integration, so your answer should be a number. fx(x) (c) What is the expected value of X? (d) What is the variance of X? (e) What is E[2X² - 3X + 1]? (f) Calculate the probability that X| is less than 2. (Your answer can be an integral, but you can also take advantage of the simple structure of the PDF.) (g) Let B = (−2,2). Determine the conditional PDF fx|Â(x) of X given the event {X € B}. (h) Calculate the probability that X < 0 given that X is less than 2. (i) Calculate the conditional expectation E[X|B] of X given that X is less than 2.

Suppose that a random variable X has the following moment generating function, < ¼/7. MX(t) = = (1-9t)−7, t< (a) Find the mean of X. (b) Find the variance of X.

Answer part a of the following question. Show work. Let Y > 0 be a continuous random variable representing time from regimen start to bone-marrow transplant. Everyone does not survive long enough to get the transplant. Let X > 0 be a continuous random variable representing time from regimen start to death. We can assume X ⊥ Y and model time to death as X ∼ Exp(rate = θ) and time to transplant as Y ∼ Exp(rate = µ). Where Exp(rate = λ) denotes the exponential distribution with density f(z | λ) = λe−λz for z > 0 and 0 elsewhere - with λ > 0. a.) Compute the probability that a patient receives transplant before they pass away (die) by integrating over the joint density fxy.

Don't Hand writing in solution.