Learning Probability: Solving a Density Function Problem

In summary, the conversation is about a person trying to learn probability and facing difficulty with a specific problem involving a density function. They also ask for an explanation of other probability problems and thank the person who provided an explanation.
  • #1
Niels
10
0
I'm trying to learn probability on my own and I'm stuck.

My multiple-variable-calculus is not so strong so the following problem got me stuck.

I have density function

f(x,y) = x^2 + xy/3 for 0<x<1; 0<y<2 otherwise 0

And I need to calculate Prob(X > Y). X and Y are random variables.

I know how to do Prob(X <= 0.5) etc.

Also would be nice if someone could explain Prob(Y < 1/2 and X < 1/2) and
Prob(X+Y < 1)

Thanks
/Niels
 
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  • #2
Niels said:
I'm trying to learn probability on my own and I'm stuck.

My multiple-variable-calculus is not so strong so the following problem got me stuck.

I have density function

f(x,y) = x^2 + xy/3 for 0<x<1; 0<y<2 otherwise 0

And I need to calculate Prob(X > Y). X and Y are random variables.

I know how to do Prob(X <= 0.5) etc.
Just to make sure, this is just integrating over all values of Y, such that X <= 0.5, which is a rectangle.

Niels said:
Also would be nice if someone could explain Prob(Y < 1/2 and X < 1/2)
Similarly, this is an integration of the density function over the region where Y < 1/2 and X < 1/2. This is a square of sidelength 1/2 with one corner at the origin. It is a double integral that can be written as:
[tex]\int_0^{\frac{1}{2}} \int_0^{\frac{1}{2}} f(x,y) dy dx[/tex]
Do you know how to find P(X<1/2 OR Y<1/2) ?
Niels said:
and
Prob(X+Y < 1)
If you cannot picture the region, rewrite it in a friendlier form; ie., Y < -X + 1. You're then integrating the density function for all points below the line y = -x + 1. This integral can be written:
[tex]\int_0^1 \int_0^{-x+1} f(x,y) dy dx[/tex]
So your original problem, P(Y<X) is just the set of points below the line y=x.
 
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  • #3
thank you!
 

FAQ: Learning Probability: Solving a Density Function Problem

What is a density function?

A density function is a mathematical representation of the probability distribution of a continuous random variable. It describes the likelihood of a random variable taking on a particular value within a given range.

What is the purpose of solving a density function problem?

Solving a density function problem allows us to calculate the probability of a continuous random variable falling within a specific range or taking on a particular value. This is important in many fields, including statistics, economics, and engineering.

What are some common methods for solving a density function problem?

Some common methods for solving a density function problem include using the cumulative distribution function, the probability density function, and the normal distribution table. Other techniques may include integration and differentiation.

How can I improve my understanding of probability and solving density function problems?

To improve your understanding of probability and solving density function problems, it is important to practice and work through a variety of problems. Reading textbooks and taking courses in probability and statistics can also help deepen your understanding.

Are there any common mistakes to avoid when solving a density function problem?

One common mistake to avoid when solving a density function problem is using the wrong formula or method. It is important to carefully read the problem and choose the appropriate formula or method based on the given information. It is also important to pay attention to units and make sure they are consistent throughout the problem.

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