# 2-Dimensional random variable probability

• tomelwood
In summary, the homework statement asks for help with a couple of questions from a past paper and the person is not sure how to proceed. They have worked out that the density function is only defined for x on [0,1] and y on [0,1] and that to find the pdf, f, and covariance, they need to integrate everywhere they are defined and set equal to one. They have also worked out that to find the marginal probabilities, they need to integrate over the range of where it is defined, with respect to the other variable. They have found that the covariance is E[(X-2)(Y-1.5)].

## Homework Statement

Hi, at the moment I am trying to revise for my Probability exam, and a couple of the questions on the past paper are as follows, however I can find nothing in our notes that is of any use! Any help would be greatly appreciated, thankyou.

i) Two random variables X and Y have density function:

f$$_{X,Y}$$(x,y) = axy$$^{2}$$ if 0$$\leq$$x$$\leq$$1, 0$$\leq$$y$$\leq$$1 and 0 otherwise
Determine the constant a such that f is a density function.

ii)Let (X,Y) be a random vector with density function:

f$$_{X,Y}$$(x,y) = 6x , if 0<x<y<1 , 0 otherwise
Compute Cov(X,Y) and $$\rho$$(X,Y) , where $$\rho$$ is the correlation.

## The Attempt at a Solution

i) For this one I'm not entirely sure what to do here. I feel I should integrate it to give me the pdf, F, but then I don't know what to do with it. What condition should it satisfy that I can impose to give me a set value for 'a'?

ii)For this, I know that the formula for Covariance is E[(X-E[X])(Y-E[Y])] and that the correlation is $$\frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}$$
So does that mean that Cov(X,Y) is just E[g(X,Y)] with g(X,Y) equalling X and Y respectively?
In which case E[X] is just the integral of the density function multiplied by x (or y) as in the 1 dimensional case?
(You may have noticed I had a moment of inspiration halfway through writing this, but have carried on as I am not sure if my inspiration is correct!)

OK Having thought about it some more I have figured out the following:
i)
since the pdf (I realize that f is the pdf, and F is the cdf, not as I wrote before) is only defined for x on [0,1] and y on [0,1] and that for the definition of pdf I need to integrate everywhere they are defined and set equal to one. So doubly integrate between 0 and 1 both times, the function ax(y^2) dxdy and set equal to one to get a value of a=6, yes?
(I now need to determine P(Y<X) , P(Y<X^2), P(max(X,Y)>=0.5) which I am now slightly lost on, though...)

ii)OK To find E[X] and E[Y] I need to know the marginal probabilities f(x) and f(y). I know the formulas for these are f(x) = integral over y (f(x,y) dy) and similar for y.
I have worked out that the region 0<x<y<1 is described by the line y=1-x (and verified that $$\int$$$$^{1}_{0}$$$$\int$$$$^{1-x}_{0}$$6x dy dx = 1, so this is right)
So now how do I find f(x) and f(y)?

If I am right in saying that to find f(x) and f(y) just integrate over the range of where it is defined, with respect to the other variable. Ie. f(x) = (integral between 1 and 0) of 6x dy = 6x
and f(y) = (integral between 1 and 0) of 6x dx = 3
Now to find E[X] I do = (integral between 0 and 1) x*6x dx = 2 ??

If this is all true, then the covariance is now E[(X-2)(Y-1.5)]. How on Earth do you work that out??

## 1. What is a 2-dimensional random variable?

A 2-dimensional random variable is a mathematical concept that represents two random variables in a single system. It is used to model the probability of outcomes in a two-dimensional space.

## 2. How is the probability of a 2-dimensional random variable calculated?

The probability of a 2-dimensional random variable is calculated by finding the area under the probability density function (PDF) curve. This can be done by integrating the PDF over the desired region.

## 3. What is the difference between a 2-dimensional random variable and a 1-dimensional random variable?

A 2-dimensional random variable takes into account two variables, while a 1-dimensional random variable only considers one. In other words, a 2-dimensional random variable has two axes (x and y), while a 1-dimensional random variable only has one axis (x).

## 4. How is a 2-dimensional random variable represented graphically?

A 2-dimensional random variable is typically represented graphically as a probability distribution or a scatter plot. The probability distribution shows the probability of different outcomes on a 2-dimensional grid, while the scatter plot displays the relationship between the two variables.

## 5. What are some real-life applications of 2-dimensional random variables?

2-dimensional random variables are commonly used in fields such as finance, engineering, and biology. They can be used to model stock prices, weather patterns, and genetic traits, among other things. They are also important in statistical analysis and decision-making processes.

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