Question

A system consists of three components A, B and C, which fails independently with probabilities 0.2, 03 and 0.2. Let X be the total number of failed components.

(a) Find the probability distribution of X.

(b) What is the probability that at least one component is working.

(c) Find E(X^3 − 1).

3. (7 points) A system consists of three components A, B and C, which fails independently with probabilities 0.2, 03 and 0.2. Let X be the total number of failed components. (a) Find the probability distribution of X. (b) What is the probability that at least one component is working. (c) Find E(X3 - 1).

Answer 1

so we have 3 components: A, B,C. 1. - X= total no. of failed components. Let PLA), P(B), Ples be the prob of A, B, C failing. now, x=0 no component fails a Plx=o) = PLĀ) PCB) PCE = 0.8) (0.7) 10.8) - = P(ABC) (as A. B,C. are indep) 0.448 x=1 exactly I component fails Plx=1) = PC Ā B C) + PLĀ B7) + P(ABC) = (0.891017) (0-2) + (0.8)(0-3)(018)+ 02) (0-7)(0.By = 0.416 P(x=2) = PLĀŠC) + P(ABC) + P(ABC) = (0.8) (0.3) (0, 2) - 0.124 + (1-2) (0-7)(0-2) + (0:2)10-31 108 P(X=3) = PIA BC) = (0-2)(0-3) (0.2) = 0.012 P(x-x) 1 0.448 10.416 10.124 10.012 (6) Plat least one component works) - PLx711) = 1- P(x=0) = 1-0.448 = 0.552 ( E(x²-1) (x²) -1 = 1.732-1= 0.732 as E (x²) x² P(X= = 0 + 0.116 + 80.124 +27+0.012 - 1734