[Mixed Random Variables] Consider a random variable, Y, which has a quasi-Bernoulli structure. With probability p ≤ [0, 1] it takes value 0, and with probability 1 – p it is described by a continuous random variable, X, with the following PDF, 1+x, x = [-1,0), x = [0, 1], otherwise. fx (x) = {1-x, (a) Obtain CDF of Y, Fy (y), and draw the sketch. (b) Find P(Y ≥ 0), P(Y > 0), and P(−½ ≤ Y ≤ ¹/). (c) Find E(Y) and V(Y). Express the results using parameter p = [0,1] (in case these depend on p).

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[Mixed Random Variables] Consider a random variable, Y, which has a quasi-Bernoulli structure.
With probability p ≤ [0, 1] it takes value 0, and with probability 1-p it is described by a continuous
random variable, X, with the following PDF,
1 + x,
fx(x)= 1- x,
0,
x = [−1,0),
x = [0, 1],
otherwise.
(a) Obtain CDF of Y, Fy(y), and draw the sketch.
3
(b) Find P(Y ≥ 0), P(Y > 0), and P(− ≤ Y ≤ ½ ).
(c) Find E(Y) and V(Y).
Express the results using parameter p € [0, 1] (in case these depend on p).
Transcribed Image Text:[Mixed Random Variables] Consider a random variable, Y, which has a quasi-Bernoulli structure. With probability p ≤ [0, 1] it takes value 0, and with probability 1-p it is described by a continuous random variable, X, with the following PDF, 1 + x, fx(x)= 1- x, 0, x = [−1,0), x = [0, 1], otherwise. (a) Obtain CDF of Y, Fy(y), and draw the sketch. 3 (b) Find P(Y ≥ 0), P(Y > 0), and P(− ≤ Y ≤ ½ ). (c) Find E(Y) and V(Y). Express the results using parameter p € [0, 1] (in case these depend on p).
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