Question

solve 7. problem 2-8.1

Announcement: Midterm is 10/28/2019 in class "Probabilistic Methods of Signal and System Analysis," G.R. Cooper and C.D. McGillem, 3rd Ed, 1999. 1. Problem 2-3.3. Change the problem to say Y=e^-X| 2. For the problem in 2-3.3, find a) the mean value of x. b) the mean square value of X. c) the Variance of X. 3. Problem 2-4.3. But change the problem to say f(y)=Ky^2 in O<y<=6. 4. Problem 2-6.2. Change to X with Gaussian with mean 1 and variance 10. 5. Problem 2-7.1. Change to say MTF of 9 years. 6. Problem 2-7.3. Change the distribution of Theta to be uniformly distributed from -pi to 3pi. 7. Problem 2-8.1. Use the changes we made to Problem 2-7.3 above. 8. Problem 2-8.2. Use the changes we made to Problem 2-7.1 above.

Answer 1

2-8.1) The time to failure of the satellite is exponentially distributed .

XEXP 1/10)

The PDF is

fx(x) ex, x > O

The CDF is

Fx(x) 1- e,x > 0

a) The conditional probability,

Р (X > 10nХ Р(X > 5) Р (X > 10) Р (X > 5) e 10/10 e-5/10 Р(X> 10X > 5) 3D е 0.5 Р (X > 10|X > 5)%3 0.6065 > 5) Р (X > 10X > 5) - Р (X> 10X> 5)%3D Р (X > 10|X> 5) 3 =

b) The conditional distribution is

e x fxix>3 (x) P(X 3X> 3 >

The conditional expectation is

​​​​​​​

Е(XX > 3) %3D xfxх>3 (х) dx 3 Ле-Аx Е(XX> 3) 3D dx х Р(X > 3) 0.1е-0.1x 3 Е(XX > 3) %3D -dx х е- 0.3 Хе -0.1x 3 Е(X\X > 3) %3D 0.1 dx хр- + 100е 0 е- 0.3 10|-хе 0.1x] Е(X\X > 3) %3 о.1- е- 0.3 30е 0.3 + 100е-0.3 Е(X\X > 3) %3D 0.1- E(X X> 3) 13 years е-0.3