Let X and Y be an arbitrary integrable random variables on (N, F, P), and let Z(w) = aX(w) + bY (w), where a, b e R. Prove formally, from first princi- ples (starting with simple functions, then non-negative, then arbitrary) the linearity

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A: It is a fact of statistics. It is widely used.

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A: Hello! As you have posted 6 different questions, we are answering the first question. In case you…

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A: We have given that a joint pdf of variables X and Y.

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A: Part (a):Here, M ~ Unif (0, 1), N ~ Expo (1/2), which are independent. Thus, the joint pdf is:

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Q: || f(x,y)dxdy = 1 if X and Y are discret random variable

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A: The joint pdf of X and Y is, fx,y=αxyI0,1xI0,x2y         =αxy, 0&lt;x&lt;1, 0&lt;y&lt;x2

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A: Step 1: Step 2: Step 3: Step 4:

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A: It is an important fact in statistics . It is a basic instance in probability distribution theory .

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