# Sum of Two Independent Random Variables

• Shannon Young
In summary: The correct density of Z = X + Y is:fZ = 1/2 if -2<z<0 or 0<z<2 and 0 otherwiseIn summary, the density of Z = X + Y is 1/2 within the intervals of -2<z<0 and 0<z<2, and 0 otherwise.
Shannon Young
Suppose X and Y are Uniform(-1, 1) such that X and Y are independent and identically distributed. What is the density of Z = X + Y?

Here is what I have done so far (I am new to this forum, so, my formatting is very bad). I know that
fX(x) = fY(x) = 1/2 if -1<x<1 and 0 otherwise

The density of Z will be given by
fZ= $$\int$$fX(z-y)fY(y)dy

fY(y) = 1/2 if -1<y<1 and 0 otherwise
So,

fZ= $$\int$$(1/2)fX(z-y)dy (bounds of integration -1 to 1)

The integrand = 0 if -1<z-y<1 or if z-1<y<z+1

That is where I get stuck, and need help to complete. Your assistance is appreciated, thanks

Shannon Young said:
Suppose X and Y are Uniform(-1, 1) such that X and Y are independent and identically distributed. What is the density of Z = X + Y?

Here is what I have done so far (I am new to this forum, so, my formatting is very bad). I know that
fX(x) = fY(x) = 1/2 if -1<x<1 and 0 otherwise

The density of Z will be given by
fZ= $$\int$$fX(z-y)fY(y)dy

fY(y) = 1/2 if -1<y<1 and 0 otherwise
So,

fZ= $$\int$$(1/2)fX(z-y)dy (bounds of integration -1 to 1)

The integrand = 0 if -1<z-y<1 or if z-1<y<z+1

That is where I get stuck, and need help to complete. Your assistance is appreciated, thanks
The integrand = 0 if -1<z-y<1 or if z-1<y<z+1
This is incorrect, it should read =1/2 within the interval and =0 outside.

mathman said:
This is incorrect, it should read =1/2 within the interval and =0 outside.

I figured the problem out, thanks

## What is the "Sum of Two Independent Random Variables"?

The sum of two independent random variables refers to the result of adding two separate random variables together. This is a common mathematical operation in statistics and probability, and it allows for the analysis and prediction of more complex systems.

## How do you calculate the sum of two independent random variables?

To calculate the sum of two independent random variables, simply add their individual values. For example, if X and Y are two independent random variables, the sum of X+Y would be equal to the sum of their individual values, denoted as X+Y.

## What is the relationship between the sum of two independent random variables and their individual distributions?

The sum of two independent random variables will have a distribution that is the convolution of their individual distributions. This means that the probability of a particular sum occurring is equal to the sum of the probabilities of the individual values that result in that sum.

## Can the sum of two independent random variables be used to predict the behavior of a more complex system?

Yes, the sum of two independent random variables can be used to predict the behavior of a more complex system. By understanding the individual distributions and relationships between the variables, the sum of the two can provide insights into the overall system.

## What are some real-world applications of the sum of two independent random variables?

The sum of two independent random variables has various applications in fields such as finance, engineering, and physics. It can be used to analyze stock market trends, model the behavior of complex systems in engineering, and predict the outcome of physical experiments involving multiple variables.

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