Sum of independent exponential distributions with different parameters

[newbiewannabe]

Homework Statement

As the title indicates. I'm given two independent exponential distributions with means of 10 and 20. I need to calculate the probability that the sum of a point from each of the distributions is greater than 30.

Homework Equations

X is Exp(10)
Y is Exp(20)

f(x) = e^(-x/10) / 10
g(y) = e^(-x/20) / 20

The Attempt at a Solution

I think I need to find the distribution of X + Y and then integrate to find the cdf. I'm not sure how to go about that though.

 

[haruspex]

What is the probability that X lies between x and x+dx? Given that X lies in that range, what is the probability that X+Y < z (for small dx)?

 

[newbiewannabe]

For small dx isn't that just the pdf of x.

P(X+Y<z) = P(X<z-Y) = F(z-Y)
I don't know how that helps though, assuming I'm correct.

 

[ehild]

What is the probability that a<x a+da and b<y<b+db?

z=x+y is a new random variable. You need the probability density f(z). The probability that z>30 comes from the cumulated distribution function F(z).

ehild

 

[Ray Vickson]

Homework Statement

As the title indicates. I'm given two independent exponential distributions with means of 10 and 20. I need to calculate the probability that the sum of a point from each of the distributions is greater than 30.

Homework Equations

X is Exp(10)
Y is Exp(20)

f(x) = e^(-x/10) / 10
g(y) = e^(-x/20) / 20

The Attempt at a Solution

I think I need to find the distribution of X + Y and then integrate to find the cdf. I'm not sure how to go about that though.

Yes, you need to find the pdf of X+Y. Just apply standard formulas for the density of a sum of independent random variables, in terms of the individual densities. If it is not in your textbook you can find all that you need on-line. Google 'sum of independent random variables'.

 

[haruspex]

For small dx isn't that just the pdf of x.

Almost, but dx must come into it.

P(X+Y<z) = P(X<z-Y) = F(z-Y)
I don't know how that helps though, assuming I'm correct.

Right idea, but you can't write F(z-Y) since Y is a random variable, not a number.
I did say "given that X lies between x and x+dx": F_X+Y(z)|X=x = P(X+ Y<z | X=x) = P(Y<z-X | X=x) = F_Y(z-x).
So, put those two together to get the probability that X+Y<z & x < X < x+dx.

 

[vela]

It also might help to consider the xy-plane. What region does the condition x+y<c represent in the plane?