# Question

Suppose the sample space Ω consists of the four points

w1, w2, w3, w4,

and the associated probabilities over the events are given by

Define the random variable X1 by

X1 (w1) = 1,

X1 (w2) = 1,

X1 (w3) = 4,

X1 (w4) = 5,

and the random variable X2 by

X1 (w1) = 1,

X2 (w2) = 1,

X2 (w3) = 1,

X2 (w4) = 5,

(a) Find the probability distribution of X1, that is, PX1(i).

(b) Find E(X1).

(c) Find the probability distribution of the random variable X1 X2, that is, PX1+ X2(i).

(d) Find E(X1 + X2) and E(X2).

(e) Find FX1X2(b1, b2).

(f) Compute the correlation coefficient between X1 and X2.

(g) Compute E [2X1 – 3X2].

w1, w2, w3, w4,

and the associated probabilities over the events are given by

Define the random variable X1 by

X1 (w1) = 1,

X1 (w2) = 1,

X1 (w3) = 4,

X1 (w4) = 5,

and the random variable X2 by

X1 (w1) = 1,

X2 (w2) = 1,

X2 (w3) = 1,

X2 (w4) = 5,

(a) Find the probability distribution of X1, that is, PX1(i).

(b) Find E(X1).

(c) Find the probability distribution of the random variable X1 X2, that is, PX1+ X2(i).

(d) Find E(X1 + X2) and E(X2).

(e) Find FX1X2(b1, b2).

(f) Compute the correlation coefficient between X1 and X2.

(g) Compute E [2X1 – 3X2].

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