# Question: Suppose the sample space consists of the four points w1

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].

## Answer to relevant Questions

Suppose that E1, E2, . . . , Em are mutually exclusive events such that E1 U E2, U . . .,Em = Ω; that is, exactly one of the E events will occur. Denote by F any event in the sample space. Note that and that FE1, i = 1, 2, ...The random variable X can take on only the values 0, ±1, ±2, and (a) Find the probability distribution of X. (b) Graph the CDF of X. (c) Compute E(X). A time-to-failure distribution is said to have a Weibull distribution if the cumulative distribution function is given by F(t) = 1 –e–tβ /ƞ , where ƞ, β > 0. Find the failure rate, and show that the Weibull ...Consider a system consisting of five components, labeled 1, 2, 3, 4, 5. The system is able to function satisfactorily as long as at least one of the following three combinations of components has every component in that ...Reconsider Prob. 17.6-32. (a) Formulate part (a) to fit as closely as possible a special case of one of the decision models presented in Sec. 26.4. (Do not solve.) (b) Describe Alternatives 2 and 3 in queueing theory terms, ...Post your question