If X and Y are continuous random variables such that Y|X = x is uniformly distributed over [x, x + 1] and X is uniformly distributed over [0, 1], find the joint pdf of X and Y (including its domain), P[X +Y < 1], and E[X|Y].

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A: (i) As if f(x) is a pdf then ∫-∞∞f(x)dx = 1⇒ ∫01axdx + ∫12adx + ∫23(-ax + 3a)dx = 1⇒ ax2201 + ax12 +…

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Q: 1. Let X and Y be two jointly continuous random variables with joint PDF 1 fxy (x, y) x+++y -1 0,Y…

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A: The answer to the above question is as follows :

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A: Solution: From the given information,

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Q: 1. Let X and Y be two jointly continuous random variables with joint pdf f(x,y) = {k(x + y), 0≤x≤…

A: Given that                      We have to find  Var(X)  And  P(1/2&lt;X&lt;1, 0&lt;Y&lt;1/2)

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Q: Suppose that X and Y are continuous random variables with CDF 0, x<0, y<0 0.5xy(x+ y) 0<x<1,0< y<1…

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A: Cumulative Distribution Function is the probability that shows the overall chance of occurrence of…