Help on the density of sum of two uniform variables.

In summary, the conversation discusses calculating the density function of Z=X+Y, where X and Y are independent uniform distributions on [0,1]. The calculation is done in two steps, with the first step being correct and the second step being incorrect due to the incorrect domain of x. The correct domain for x is z-1 to 1, resulting in a corrected density function of f(z)=2-z. The conversation ends with gratitude for the hint provided by the expert summarizer.
  • #1
gimmytang
20
0
Hi, I need to calculate the density function of Z=X+Y, where X and Y are independent uniform distributed on [0,1]. The calculation is in the following:
[tex]f_{Z}(z)=\int_{A}dx[/tex]
a. If 0<z<1, A={x:0<x<z} then f(z) = z;
b. If 1<z<2, A={x:0<x<1} then f(z) = 1;
Step b is wrong, but I don't know where I am wrong. Any hint will be appreciated!
Thanks
gim :cry:
 
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  • #2
For step b, the domain of x is z-1 to 1, so f(z)=2-z.

The reason for that is z-x=y, which is restricted to (0,1).
 
  • #3
yep, you are right. thanks!
 

1. What is the definition of density in the context of sum of two uniform variables?

The density of sum of two uniform variables refers to the probability distribution of the sum of two random variables that are uniformly distributed. It describes the likelihood of obtaining a particular sum of two values from the two variables.

2. How is the density of sum of two uniform variables calculated?

The density of sum of two uniform variables is calculated by convolving the two individual uniform density functions. This involves integrating the product of the two density functions over all possible values of the sum.

3. Can the density of sum of two uniform variables be negative?

No, the density of sum of two uniform variables cannot be negative. It is always non-negative, meaning it can take on values of zero or positive numbers. This is because probabilities cannot be negative.

4. What is the relationship between the density of sum of two uniform variables and the individual uniform density functions?

The density of sum of two uniform variables is the convolution of the two individual uniform density functions. This means that it takes into account the probabilities of both individual variables and combines them to give the overall probability of obtaining a certain sum.

5. How is the density of sum of two uniform variables used in practical applications?

The density of sum of two uniform variables is used in various fields such as statistics, engineering, and physics to model and analyze real-world phenomena. It is also used in simulations and statistical tests to make predictions and draw conclusions about the sum of two random variables.

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