# Sum of independent exponential distributions with different parameters

• newbiewannabe
In summary: What is the probability density of the sum of two independent random variables in the plane?In summary, the task is to find the probability that the sum of a point from two independent exponential distributions, with means of 10 and 20, is greater than 30. To do this, we need to find the distribution of X+Y and then integrate to find the cumulative distribution function (cdf). This can be done by applying standard formulas for the density of a sum of independent random variables. We also need to consider the xy-plane and find the region that represents the condition of x+y<c in the plane. The probability density of the sum of two independent random variables in the plane will help us calculate the overall probability.
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.

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)?

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.

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

newbiewannabe said:

## 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'.

newbiewannabe said:
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": FX+Y(z)|X=x = P(X+ Y<z | X=x) = P(Y<z-X | X=x) = FY(z-x).
So, put those two together to get the probability that X+Y<z & x < X < x+dx.

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

## 1. What is the sum of independent exponential distributions with different parameters?

The sum of independent exponential distributions with different parameters is a mathematical concept that refers to the combined probability distribution of a set of independent random variables, each following an exponential distribution with its own unique parameter.

## 2. What is the significance of studying this concept?

Studying the sum of independent exponential distributions with different parameters is important in various fields of science, such as statistics, physics, and engineering. It helps in understanding the behavior and properties of complex systems, as well as in making predictions and calculations based on probability distributions.

## 3. How is this concept related to the Central Limit Theorem?

The Central Limit Theorem states that the sum of a large number of independent random variables, regardless of their underlying distributions, tends to follow a normal distribution. The sum of independent exponential distributions with different parameters is a special case of this theorem, where the random variables follow exponential distributions.

## 4. Can the sum of independent exponential distributions with different parameters have non-exponential distributions?

Yes, in some cases, the sum of independent exponential distributions with different parameters can result in non-exponential distributions, such as the gamma distribution. This happens when the parameters of the exponential distributions are not identical.

## 5. What are some real-world applications of this concept?

The sum of independent exponential distributions with different parameters has various applications in fields such as finance, biology, and telecommunications. It can be used to model the time between events, such as the arrival of customers at a store, the occurrence of mutations in DNA sequences, and the transmission of data packets in a network.

Replies
7
Views
538
Replies
5
Views
2K
Replies
2
Views
1K
Replies
13
Views
2K
Replies
4
Views
1K
Replies
4
Views
1K
Replies
3
Views
6K
Replies
7
Views
1K
Replies
5
Views
2K
Replies
1
Views
967