Question

There are two fuses in an electrical device. Let X denote the lifetime of the first fuse, and let y denote the lifetime of the second fuse both in years). Assume the joint probability density function of X and Yis f(x,y) – $(x +2y). 0<x<1, 0 <y<2 a. What is the probability that both uses last longer than 4 months? b. What is the probability that the second fuse lasts less than 3 months given that the first fuse last 6 months? c. Find the expected lifetime of the second fuse given that the first fuse last 6 months. d. Find the expected lifetime of the first fuse. e. Are X and Y independent? (write independent or not independent" here.Please explain the details on your paper.)

Answer 1

(a)

4 months = 4 /12 years = 1/6 years

The requied probability is

$P(X>1/3,Y>1/3)=\int_{1/3}^{1}\int_{1/3}^{2}f(x,y)dydx=\frac{1}{5}\int_{1/3}^{1}\int_{1/3}^{2}(x+2y)dydx$

5.C 35 - S, [ry + y*) , dr = SE 2.c +4 cole dc ALL + dc 3 3 9

$=\frac{1}{5}\left [ \frac{5x^{2}}{6}+\frac{35x}{9} \right ]_{1/3}^{1}=\frac{1}{5}\left [ \frac{5}{6}+\frac{35}{9}-\frac{5}{54}-\frac{35}{24} \right ]=0.6343$

(b)

The marginal pdf of first fuse, X is

f(a) = 5(x + 2y)dy [ry +4*1= x + 2) 5

The marginal pdf of second fuse Y is

$f(y|x)=\frac{f(x,y)}{f(x)}=\frac{\frac{1}{5}(x+2y)}{\frac{2}{5}(x+2)}=\frac{x+2y}{2(2+x)}$

-------------------

6 months = 0.5 years

3 months = 0.25 years

Now for X= 0.5 we have

$f(y|X=0.5)=\frac{0.5+2y}{2(2+0.5)}=\frac{1}{10}(1+4y)$

The requried probability is

$f(Y<0.25|X=6)=\frac{1}{10}\int_{0}^{0.25}(1+4y)dy=\frac{1}{10}\left [ y+2y^{2} \right ]_{0}^{0.25}=0.0375$

(c)

$E(Y|X=0.5)=\int_{0}^{2}yf(y|X=0.5)dy=\frac{1}{10}\int_{0}^{2}(y+4y^{2})=\frac{1}{10}\left [ \frac{y^{2}}{2}+\frac{4y^{3}}{3} \right ]_{0}^{2}\approx1.27$

(d)

$E(X)=\int_{0}^{1}xf(x)dx=\int_{0}^{1}\frac{2}{5}(x^{2}+2x)dx=\frac{2}{5}\left [\frac{x^{3}}{3}+x^{2} \right ]_{0}^{1}=0.5333$

(e)

The marginal pdf of Y is

$f(y)=\int_{0}^{1}\frac{1}{5}(x+2y)dx=\frac{1}{5}\left [ \frac{x^{2}}{2}+2xy \right ]_{0}^{1}=\frac{1}{5}\left [\frac{1}{2}+2y \right ]$

Since

$f(x,y)\neq f(x)f(y)$

So X and Y are not independent.