(a) Let Y be a continuous random variable with distribution Uniform(a, b), for some a

(a) Let Y be a continuous random variable with distribution Uniform(a, b), for some a < b. Find the c.d.f. of Y. (b) Now let Xl be Uniform(0, 2), X2 be Uniform(l, 4) and Z be Bernoudlli(p), for some 0 < p < 1. We are interested in the random variable X, defined as XI if Z = 1 x ifZ=O. You may think of X as being obtained in the following way: you flip a coin having probability p of landing heads. If the outcome of the coin flip is heads, you get to observe Xl, otherwise you observe X2. The random variable X is called a mixture of Xl and X2 with mixing probability p. Show that the cdf of X is Fx@) = pFXl@) + (1 - Hint: use the law of total probability. (c) Using the previous result show that the pdf of X is P 2 fx@) 3