Question

Question 3: Two components in a personal computer system have lifetimes (in years) that are distributed with a joint probability density function given by fx.Y(,y e-#(1+v), 0 < r,0 y, 0, otherwise (a) (3 marks). Find fx(x) (b) (3 marks). Find fy(y) (c) (3 marks). Find the probability that the lifetime of at least one com ponent exceeds 2 years. Question 4

ALL OF QUESTION 3

Answer 1

Answer:

Given that:

(a) Find
fx(x)

The marginal pdf of X is

f (π,y) dy fx(x)

xe(1+y)d

rydy = re -r e

e hr- re -r

  

e

(b)  Find $f_{Y}(y)$

The marginal pdf of Y is

$f_{Y}(y)=\int_{0}^{\infty}f(x,y)dx$

$=\int_{0}^{\infty}xe^{-x(1+y)}dx$

  

yx 1ez{y+1) 7] (y1)2

  $=\frac{1}{(1+y)^{2}}$

(c) Find the probability that the lifetime of at least one component exceeds 2 years.

Let us first find the probability that the lifetime of both components is less than 2 years. So

$P(x<2,y<2)=\int_{0}^{\infty}\int_{0}^{\infty}f(x,y)dydx$

$=\int_{0}^{\infty}\int_{0}^{\infty}xe^{-x(1+y)}dydx$

$=\int_{0}^{2}\left [-e^{-x(1+y)} \right ]_{0}^{2}dx$

$=\int_{0}^{2}\left [e^{-x}-e^{-3x} \right ]dx$

$=\left [-e^{-x}+\frac{e^{-3x}}{3} \right ]_{0}^{2}$

$=0.5322$

So required probability is

PX 2 or Y> 2) 1-0.5322 0.4678