Joint Distribution Puns

I am learning about continuous joint distributions and marginal densities - is there anyone able to help me understand the differences between joint CDF and marginal densities?

I thought that marginal densities are the same thing as finding the CDF of Y (or X) for a joint distribution f(x, y). Are these not the same?

I understand, for example, that the marginal density of y is equal to h(y), which is the integral of f(x,y) with respect to x. If marginal densities aren't the same, then what is the CDF of only one variable for a continuous joint distribution?

X = {0,1} ; Y={0,1,2}

I want to find a, b & c s.t X and Y are independent. Mostly I am concerned with finding marginal if that leads to finding a,b,c.

I have found following things I need to find the marginals of Y.

p_X(0) = 3/4

p_X(1) = 1/4

------------------------------------------------------------------------------------------------------------------------------------

p_Y(0) = a + 1/8

p_Y(1) = 1/2

p_Y(2) = b+c

So recently I was playing around with sampling distributions of elements of some rank -- for example, given n applicants to a given job, what's the distribution of top applicants look like, relative to the broader population distribution from which they're drawn? The second best applicants? The third? etc. Turns out the marginal distribution of elements of rank k is Beta(n+1-k, k) distributed, which I confirmed with a quick simulation in R, first varying n and then varying k. These quantiles can then be transformed back into the original scale via the population distribution's quantile function.

Suppose I want to jointly sample the top two candidates in a similar manner. I can't simply draw each from their marginal distributions, as that would not respect their order (i.e. the best candidate must by construction correspond to a higher quantile than the second best). What sort of distribution describes these quantiles? I tried thinking of ways to shoehorn the dirichlet in here but ultimately came up with nothing (since there doesn't seem to be any necessity for the quantiles or the differences between them to form a simplex).

I was shocked to see "13-year-old who shared pot cigarette charged with felony" in Tuesday's edition of the Grand Island Independent newspaper. The child allegedly admitted to sharing a joint with others on the way to school one morning. If true, this is certainly abhorrent and the child deserves tough corrective intervention.

But Police Capt. Jim Duering justifying charging the boy with a Class III Felony by saying "[the statute] doesn't specify that you have to sell it" in order to constitute an intent to distribute seems excessive and appears to be an overreach outside the spirit of this law. I'm no attorney nor an expert on criminal charges, but. COME. ON. Joint-passing qualifies as "distributing?" Really? It seems like this statute is meant to be applied to drug dealers, not the misguided misconduct of middle schoolers.

Maybe there is relevant contextual info I don't know. I am admittedly operating on limited information. But our country's views on marijuana are changing fast: Two-thirds of Americans now support its legalization, according to Pew Research. Meanwhile more states bordering Nebraska have legalized its recreational use. If that ever happens in Nebraska or nationally (and it may soon) what these kids allegedly did before school will be more akin to a teenager breaking into their parents' liquor cabinet and sharing some of the spoils with their friends than equivalent to endeavoring to sell meth or heroin.

I'm not excusing this boy's misbehavior. Like underage drinking, underage marijuana use should be punished, especially as egregious as someone this young and before school. But numerous studies have shown that sending youth into the juvenile criminal justice system creates a cycle that is very tough to break. Let's not go out of our way to do so and instead use this as an opportunity to steer a youngster onto a better path.

Hi there! Looking for some help with this one:

https://imgur.com/a/Rlgqa2c

All help is appreciated : )

Let a pair of die is rolled. Assume that X denotes the smallest value and Y denotes denotes the largest values. Find joint mass function of X and Y.

I understand that when x=y, then the joint function will be f(x, y) = 1/36, where x=y=1, 2, 3 , 4, 5, 6. But when x and y are different ( according to the condition x<y), then how f(x, y) = 2/36? Could anyone please explain how we get 2/36 for each order pair where x is smaller than y?

Hi, I'm struggling with a homework problem. It was explained to me in office hours but I'm still struggling with the reasoning. The problem is:

Let X and Y be jointly absolutely continuous random variables. Suppose X ∼ Exponential(2) and that P(Y > 5 | X = x) = e^(−3x) . Compute P(Y > 5).

Ok. So in the chapter we have a result that f_X|Y(x|y) = f_X,Y(x,y) / f_X(x) . This doesn't apply because we have P(Y>5|X=x) (not P(Y=y|X=x)). But I don't know how else to think about this. In office hours I was told to integrate over all x the product P(Y>5|X=x) * f_X(x). Since X ~Exponential I integrated from 0 to infinity (the range for exponential). I got a constant, 2/5. Is this P(Y>5) or is this f_Y(y) ? I thought it was the latter and thought I needed to integrate again but maybe not. The mechanics are easy but I think I don't understand how this is an extension of the law of total probability. I'm just looking for this to be corrected/explained in some other way so I can understand it better. Thanks!

I am thinking that X here is discrete whereas N is continuous so the probability of X here is 1/3. I'm probably going to apply conditional probability here to solve for the probability of error. My problem is, I don't know how to represent Y = X + N and find their limits.

https://preview.redd.it/24fdmkqy89u61.png?width=1227&format=png&auto=webp&s=e569a835dd6de31eb8d9790950e6509f52a58f11

https://imgur.com/a/fUOOnVD

For this question, I do not understand why Px,y(1,0) is evaluated to be 0. The way I am looking at it, is that this is suppose to represent the probability that Chris does not make his first shot, therefore, it would evaluate to be 2/3, the complement of 1/3. Similarly, this my thought process for the other probabilities that are also evaluating to 0.

Any help would be appreciated.

Please help me with this question

Let X and Y be random variables with joint pdf: fXY(x,y)=1/4, -1<=x,y<=1; 0 otherwise. Determine P(X^2 + Y^2 < 1)

Thank you.

This is a general probability question, but I am having a lot of trouble understanding joint distributions, CDF's, and marginal distributions.

For continuous joint distributions, I thought that marginal densities are the same thing as finding the CDF of Y (or X) for a joint distribution f(x, y). Are these not the same?

I understand, for example, that the marginal density of y is equal to h(y), which is the integral of f(x,y) with respect to x. If marginal densities aren't the same, then what is the CDF of only one variable for a continuous joint distribution?

Let a pair of die is rolled. Assume that X denotes the smallest value and Y denotes denotes the largest values. Find joint mass function of X and Y.

I understand that when x=y, then the joint function will be f(x, y) = 1/36, where x=y=1, 2, 3 , 4, 5, 6. But when x and y are different ( according to the condition x<y), then how f(x, y) = 2/36? Could anyone please explain how we get 2/36 for each order pair where x is smaller than y? Thanks