# Probability Density Function - Need Help

• vptran84
In summary, the problem is to find the PDF of W = X + Y when X and Y have the joint PDF fx,y (x,y) = 2 for 0<=x<=y<=1, and 0 otherwise. After analyzing the integral and using double integration, it is found that fw(w) = w for 0<=w<=1, fw(w) = 2-w for 1<=w<=2, and fw(w) = 0 otherwise.
vptran84
Probability Density Function -- Need Help!

Hi,

Can someone please check my work if i did the problem correctly? thanks in advance.

Here is the problem:

Find the PDF of W = X + Y when X and Y have the joint PDF fx,y (x,y) = 2 for 0<=x<=y<=1, and 0 otherwise.

here is my solution:
$$\int_{0}^{1} \int_{0}^{w-y} 2dxdy$$

I work through the integral and get fw (w) = 2w-1 for w>0, and 0 for w<0.

Last edited:
Your answer is obviously wrong. f(w) is <0 for w<1/2. Moreover, the integral should be 1 - yours is 0.

ok, i did a little more thinking :tongue2: and this is what i got now...

For region w>0, the region of integration is outside so CDF Fw (w) is 0

For region 0<=w<=1, i used double integration, and i get w^2/2

For region 1<=w<=2, i get 2w-1-w^2/2

For region w>2, i get 1.

So to find PDF, i take the derivative, and i get the following:

fw(w) = w for 0<=w<=1
fw(w) = 2-w for 1<=w<=2
fw(w) = 0 otherwise.

Please let me know if i did anything wrong.

Before I looked at your latest post, I worked it out myself. I got the same result as you did.

## 1. What is a Probability Density Function (PDF)?

A Probability Density Function (PDF) is a function that describes the probability of a random variable taking on a certain value within a given range. It is used to model continuous random variables and is often represented graphically as a curve on a probability distribution.

## 2. How is a PDF different from a Probability Mass Function (PMF)?

A Probability Mass Function (PMF) is used to describe the probability of a discrete random variable taking on a specific value. This means that the PMF assigns probabilities to individual points on a distribution, while the PDF assigns probabilities to ranges of values.

## 3. What is the relationship between a PDF and a Cumulative Distribution Function (CDF)?

The Cumulative Distribution Function (CDF) is the integral of the PDF, and it represents the probability of a random variable being less than or equal to a certain value. In other words, the CDF is the sum of all probabilities up to a specific value on the distribution curve.

## 4. How do you calculate the area under a PDF curve?

The area under a PDF curve represents the total probability of all possible outcomes. To calculate this, you can use integration techniques to find the definite integral of the PDF function within a given range. Alternatively, you can use numerical methods such as the trapezoidal rule or Simpson's rule to estimate the area.

## 5. How are PDFs used in statistics and data analysis?

PDFs are used in statistics and data analysis to model and analyze continuous random variables. They are useful for calculating probabilities, determining the likelihood of certain outcomes, and understanding the distribution of data. PDFs are also used in hypothesis testing, regression analysis, and other statistical techniques.

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