X and Y are the lifetimes of two computers in an office. X and Y are independent and identically distributed variables with the joint density given below. What is the probability that both computers will become inoperable within the average lifetime? $$f(x, y) = \frac{xy}{81}exp(-\frac{x+y}{3})U(x)U(y)$$
Added by Ismael M.
Close
Step 1
Then $du = dy$ and $v = -3exp(-\frac{y}{3})$. $$\int_{0}^{\infty} y exp(-\frac{y}{3}) dy = [-3y exp(-\frac{y}{3})]_{0}^{\infty} + \int_{0}^{\infty} 3 exp(-\frac{y}{3}) dy$$ $$= 0 + [-9 exp(-\frac{y}{3})]_{0}^{\infty} = 0 - (-9) = 9$$ So, $$f_X(x) = Show more…
Show all steps
Your feedback will help us improve your experience
Adi S and 100 other Intro Stats / AP Statistics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Two components of a laptop computer have the following joint probability density function for their useful lifetimes X and Y (in years): 1.6xe^(-1.6x(1+1.6y)) x > 0, y > 0, f(x,y) = 0 otherwise. a: Find the marginal probability density function of X, f(x). Enter a formula below. Use * for multiplication, / for division, ^ for power, and exp for exponential function. For example, 3*x^3*exp(-x/3) means 3x^3e^(-x/3). f(x) = b. Find the marginal probability density function of Y, f(y). Enter a formula below. f(y) = c. What is the probability that the lifetime of at least one component exceeds 1 year (when the manufacturer's warranty expires)? Round your answer to 4 decimal places.
Adi S.
Two components of a laptop computer have the following joint probability density function for their useful lifetimes X and Y (in years): f(x,y) = { 3xe^{-x(1+3y)} x >= 0, y >= 0; 0 otherwise a. Find the marginal probability density function of X, fX(x). Enter a formula below. Use * for multiplication, / for division, ^ for power and exp for exponential function. For example, 3*x^3*exp(-x/3) means 3x^3e^{-x/3}. fX(x) = , x >= 0. b. Find the marginal probability density function of Y, fY(y). Enter a formula below. fY(y) = , y >= 0. c. What is the probability that the lifetime of at least one component exceeds 1 year (when the manufacturer's warranty expires)? Round your answer to 4 decimal places.
33. Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y: f(x,y) = {xe^-x(1+y) x ≥ 0 and y ≥ 0, 0 otherwise a. What is the probability that the lifetime X of the first component exceeds 3? b. What are the marginal pdf's of X and Y? Are the two lifetimes independent? c. What is the probability that the lifetime of at least one component exceeds 3?
Dominador T.
Recommended Textbooks
Elementary Statistics a Step by Step Approach
The Practice of Statistics for AP
Introductory Statistics
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD