Joint Probability Density Function

In summary, to find P(X + 2*Y <= 3), you need to draw the region represented by X+2Y < 3 and then integrate the joint density function on that region, which is a triangle. The value of k can be found to be lambda^2. Initially, there were some uncertainties about the solution, but it was clarified that Y = 0 and the problem can be simplified to P(X<= 3), which is easier to solve.
  • #1
Hashmeer
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0

Homework Statement


X and Y are random variables with the joint density:

fXY(x,y) = k*e^(-lambda * x) if 0 < y < x < infinity
= 0, otherwise

Find P(X + 2*Y <= 3)

Homework Equations



I found k = lambda^2



The Attempt at a Solution


I'm not sure exactly how to solve this, but here are the ideas that I was starting to work off of:

1) Y = 0 since there is no function for y and you cannot solve for it in the original equation as it does not appear there.
2) If the above is true then the problem simplifies to P(X<= 3) which is a fairly simple thing to solve for.

I'm not sure if these are correct and have no way of checking my work, so I was hoping someone could confirm my thoughts or point me in the right direction.

Thanks!
 
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  • #2
Draw the region represented X+2Y < 3 and then integrate your pdf on that region. The region is some sort of triangle.
 
  • #3
Ah, right. Thanks for the help, that was pretty dumb...
 

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

A joint probability density function (PDF) is a mathematical function that describes the likelihood of two or more random variables occurring simultaneously. It is used to model the joint behavior of multiple variables and is typically denoted as f(x,y) or p(x,y).

2. How is a joint PDF different from a regular PDF?

A regular PDF only describes the probability distribution of a single random variable, whereas a joint PDF describes the probability distribution of multiple variables together. A joint PDF takes into account the probabilities of all possible combinations of values for the variables.

3. What is the difference between a joint PDF and a joint cumulative distribution function (CDF)?

A joint PDF gives the probability of a specific combination of values for the variables, while a joint CDF gives the probability that the variables are less than or equal to certain values. In other words, a joint CDF is the cumulative sum of the joint PDF.

4. How is a joint PDF used in statistical analysis?

A joint PDF is used to calculate the probability of specific outcomes for multiple variables, which is useful in understanding the relationship between the variables and making predictions. It is also used in calculating joint moments, correlations, and other statistical measures.

5. Can a joint PDF have more than two variables?

Yes, a joint PDF can have any number of variables, not just two. For example, in a three-dimensional space, the joint PDF would be denoted as f(x,y,z) or p(x,y,z), and it would describe the probability of three variables occurring together. However, as the number of variables increases, the complexity of the joint PDF also increases.

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