Question

Problem 0.1 Suppose X and Y are two independent exponential random variables with respective densities given...

Problem 0.1

Suppose X and Y are two independent exponential random variables with respective densities given by(λ,θ>0) f(x) =λe^(−xλ) for x>0 and g(y) =θe^(−yθ) for y>0.

(a) Show that Pr(X<Y) =∫f(x){1−G(x)}dx {x=0, infinity] where G(x) is the cdf of Y, evaluated at x [that is,G(x) =P(Y≤x)].

(b) Using the result from part (a), show that P(X<Y) =λ/(θ+λ).

(c) You install two light bulbs at the same time, a 60 watt bulb and a 100 watt bulb. The lifetime of the 60 watt bulb has an exponential density with an average lifetime 1500 hours. The 100watt bulb also has an exponential density with average lifetime of 1000 hours. What is the probability that the 100 watt bulb will outlast the 60 watt bulb?

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let X and Y be independent exponential random variables with respective parameters 2 and 3. a)....
Let X and Y be independent exponential random variables with respective parameters 2 and 3. a). Find the cdf and density of Z = X/Y . b). Compute P(X < Y ). c). Find the cdf and density of W = min{X,Y }.
Assume that X and Y are independent random variables, each having an exponential density with parameter...
Assume that X and Y are independent random variables, each having an exponential density with parameter λ. Let Z = |X - Y|. What is the density of Z?
If X and Y are independent exponential random variables, each having parameter λ  =  4, find...
If X and Y are independent exponential random variables, each having parameter λ  =  4, find the joint density function of U  =  X + Y  and  V  =  e 3X. The required joint density function is of the form fU,V (u, v)  =  { g(u, v) u  >  h(v), v  >  a 0 otherwise (a) Enter the function g(u, v) into the answer box below. (b) Enter the function h(v) into the answer box below. (c) Enter the value...
If X and Y are independent exponential random variables, each having parameter λ  =  5, find...
If X and Y are independent exponential random variables, each having parameter λ  =  5, find the joint density function of U  =  X + Y  and  V  =  e 6X. The required joint density function is of the form fU,V (u, v)  =  { g(u, v) u  >  h(v), v  >  a 0 otherwise (a) Enter the function g(u, v) into the answer box below. (b) Enter the function h(v) into the answer box below. (c) Enter the value...
If X and Y are independent exponential random variables, each having parameter λ  =  4, find...
If X and Y are independent exponential random variables, each having parameter λ  =  4, find the joint density function of U  =  X + Y  and  V  =  e 9X. The required joint density function is of the form fU,V (u, v)  =  { g(u, v) u  >  h(v), v  >  a 0 otherwise (a) Enter the function g(u, v) into the answer box below. (b) Enter the function h(v) into the answer box below. (c) Enter the value...
7.5) If X1 and X2 are independent random variables having exponential densities with the parameters θ1...
7.5) If X1 and X2 are independent random variables having exponential densities with the parameters θ1 and θ2, use the distribution function technique to find the probability density of Y=X1+X2 when a) θ1 ≠ θ2 b) θ1 = θ2 7.7) With reference to the two random variables of Exercise 7.5, show that if θ1 = θ2 = 1, the random variable Z1=X1/(X1 + X2) has the uniform density with α=0 and β=1.                                      (I ONLY NEED TO ANSWER 7.7)
7.5) If X1 and X2 are independent random variables having exponential densities with the parameters θ1...
7.5) If X1 and X2 are independent random variables having exponential densities with the parameters θ1 and θ2, use the distribution function technique to find the probability density of Y=X1+X2 when a) θ1 ≠ θ2 b) θ1 = θ2 7.7) With reference to the two random variables of Exercise 7.5, show that if θ1 = θ2 = 1, the random variable Z1=X1/(X1 + X2) has the uniform density with α=0 and β=1.                                      (I ONLY NEED TO ANSWER 7.7)
Suppose X and Y are continuous random variables with joint density function f(x,y) = x +...
Suppose X and Y are continuous random variables with joint density function f(x,y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. (a). Compute the joint CDF F(x,y). (b). Compute the marginal density for X and Y . (c). Compute Cov(X,Y ). Are X and Y independent?
9. Suppose X and Y are continuous random variables with joint density function f(x,y) = x...
9. Suppose X and Y are continuous random variables with joint density function f(x,y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. (a). Compute the joint CDF F(x,y). (b). Compute the marginal density for X and Y . (c). Compute Cov(X,Y ). Are X and Y independent?
Suppose that the joint probability density function of the random variables X and Y is f(x,...
Suppose that the joint probability density function of the random variables X and Y is f(x, y) = 8 >< >: x + cy^2 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 0 otherwise. (a) Sketch the region of non-zero probability density and show that c = 3/ 2 . (b) Find P(X + Y < 1), P(X + Y = 1) and P(X + Y > 1). (c) Compute the marginal density function of X and Y...