# Help on to find probability density function

In summary, the conversation discusses finding the probability density function of a given equation, where x and y are independent uniform distributions. The group discusses using the joint distribution of x and y to find the pdf, and suggests finding the cdf first. They also mention finding the region for which the equality holds and using that to solve the integral. The conversation ends with the group suggesting to write the equation in terms of y and then using that to continue with solving the problem.
hey guys, i am really confused on something.here is the thing:
i have;

i=x+(x^2-y)^(1/2)

and here x is uniform distribution on (a,b)
y is uniform distribution on (c,d)
x and y independent
i need to find the probability density function of i but how?
actually i don't know how to start!

First you will need to know the joint distribution of X and Y. This is easy because of independence:

$$f_{X,Y}(x,y)=f_X(x)f_Y(y)$$

Now, to find the pdf, you will need to find the cdf first. That is, for each a, you will want to calculate

$$P\{X+\sqrt{X^2-Y}\leq a\}=\iint_{\{(x,y)~\vert~x+\sqrt{x^2-y}\leq a\}}{f_{X,Y}(x,y)dxdy}$$

To evaluate this integral, you'll need to know the region $\{(x,y)~\vert~x+\sqrt{x^2-y}\leq a\}$ somewhat better.

So I suggest you first find out for which tuples the equality holds. That is, for which x and y does it hold that

$$x+\sqrt{x^2-y}=a$$

(hint: the answer will be a straight line!)

hi micromass
thanks for the help but can you solve it? because i used your help to solve it but i couldn't do it.

Well, first you're going to need to write

$$x+\sqrt{x^2-y}=a$$

in function of y. What do you get for that?

ok.i got y=2ax-x^2 then what? how am i going to use this?

## 1. What is a probability density function (PDF)?

A probability density function (PDF) is a mathematical function that describes the probability distribution of a continuous random variable. It specifies the likelihood of a random variable taking on a specific value within a given range of values.

## 2. How is a PDF different from a probability mass function (PMF)?

A PDF is used to describe the probability distribution of a continuous random variable, while a PMF is used for discrete random variables. This means that a PDF can take on any value within a range, while a PMF only takes on specific values.

## 3. How do I find the PDF of a given data set?

The PDF of a data set can be found by plotting the data on a graph and then using a curve-fitting method to fit a curve to the data. This curve represents the PDF of the data set.

## 4. What is the area under a PDF curve?

The area under a PDF curve represents the probability of a random variable falling within a specific range of values. The total area under the curve is equal to 1, which signifies that the probability of the random variable taking on any value within the entire range is 1.

## 5. How is the PDF used in statistical analysis?

The PDF is used in statistical analysis to determine the likelihood of a particular outcome occurring in a given sample. It is also used to calculate important statistical measures, such as mean, variance, and standard deviation.

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