61 votes

Shouldn't the joint probability of 2 independent events be equal to zero?

There is a difference between independent events: $\mathbb P(A \cap B) =\mathbb P(A)\,\mathbb P(B)$, i.e. $\mathbb P(A \mid B)= \mathbb P(A)$ so knowing one happened gives no information about ...
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18 votes
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intuitive difference between joint probability and conditional probability in this example

You actually had your answer right there. $P(H=hit)$ is the marginal probability. It reads "The probability of getting hit.". It is the proportion of people that got hit crossing the street, ...
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  • 316
16 votes
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Proof that joint probability density of independent random variables is equal to the product of marginal densities

By definition, the random variables $X_1,\dots,X_n$ are independent iff $$ \Pr(X_1\in B_1,\dots,X_n\in B_n) = \Pr(X_1\in B_1)\dots\Pr(X_n\in B_n) $$ for every choice of Borel sets $B_1,\dots,B_n$. ...
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  • 22.1k
14 votes

Why don't we see Copula Models as much as Regression Models?

The first and most important reason is that standard regression models had a one to two-hundred year headstart on copula models (depending on exactly where you count the genesis of regression models ...
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  • 96.7k
13 votes

How to find marginal distribution from joint distribution with multi-variable dependence?

As you correctly pointed out in your question $f_{Y}(y)$ is calculated by integrating the joint density, $f_{X,Y}(x,y)$ with respect to X. The critical part here is identifying the area on which you ...
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13 votes

Conditional Expectation E[X] = E[X|Y<=a] + E[X|Y>a]

Not quite, if we use the law of total expectation we would have that $$ E(X) = E(X| Y \le a)P(Y \le a) + E(X|Y > a) P(Y>a) $$
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  • 517
13 votes

Shouldn't the joint probability of 2 independent events be equal to zero?

What I understood from your question, is that you might have confused independent events with disjoint events. disjoint events: Two events are called disjoint or mutually exclusive if they cannot ...
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11 votes
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Minimum CDF of random variables

Let $x$ by any number. Consider the event $\min(X,Y)\le x$. It can be expressed as the union of two events $$\min(X,Y)\le x = (X\le x) \cup (Y \le x),$$ shown by the overlapping yellow and green ...
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  • 290k
11 votes
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Joint distribution in layman's terms

As a concrete example, suppose I toss a coin and roll a die one after the other. As you know, there is a probability distribution associated with the outcomes of both (discrete uniform distributions, ...
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  • 1,118
11 votes
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What is the number of parameters needed for a joint probability distribution?

It takes $3\times 2 \times 2 \times 3 = 36$ numbers to write down a probability distribution on all possible values of these variables. They are redundant, because they must sum to $1$. Therefore ...
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  • 290k
10 votes
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How to find conditional distributions from joint

Those distributions you call "marginal" are not marginal. They are conditional distributions because you wrote $x \mid y$. The marginal distribution of $X$, for example, is necessarily independent ...
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  • 5,036
10 votes

What is the moment of a joint random variable?

There isn't a "the" with respect to moments, since there are many of them, but moments of bivariate variables are indexed by two indices, not one. So rather than $k$-th moment, $\mu_k$ you have $(j,k)...
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  • 261k
10 votes
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Independence and Order Statistics

Here is a guide to solving this problem (and others like it). I use simulated values to illustrate, so let's begin by simulating a large number of independent realizations from the distribution with ...
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  • 290k
9 votes
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Probability of two random variables being equal

The probability that $X_1=X_2$ is the probability that both are zero, plus the probability that both are one, plus the probability that both are two, and so on.
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9 votes
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What does it mean to factor a joint distribution?

If I am understanding the passage and table correctly, there are essentially two answers: mathematical and qualitative. Qualitatively, the "different aspects of forecast quality" is essentially the ...
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  • 12.1k
9 votes
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Maximum of a probability vector distributed as a Dirichlet variate

I am not sure there is a closed-form solution for the distribution of $p_{(k)}$ when $(p_1,\ldots,p_k)\sim\text{Dir}(\alpha_1,\ldots,\alpha_k)$ and the $\alpha_i$'s are different. At least, it ...
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  • 92.6k
9 votes
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Distribution given sum

It can be instructional and satisfying to work this out using basic statistical knowledge, rather than just doing the integrals. It turns out that no calculations are needed! Here's the circle of ...
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  • 290k
9 votes

How to make random draws from an unspecified distribution?

Presumably $w_1+w_2=1$ and $w_1,w_2 \geq 0$, so $f$ is a convex combination of $f_1,f_2$ and therefore a valid distribution (a mixture of $f_1,f_2$). Generate a Bernoulli($w_1$) random variable (i.e. ...
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  • 616
9 votes

Product of 2 Uniform random variables is greater than a constant with convolution

Some hints: Geometrical approaches are much easier for uniform RVs, but the general approach is to integrate the joint PDF in the region that satisfy $XY>\alpha$. The integral will basically look ...
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  • 52.2k
9 votes

Product of 2 Uniform random variables is greater than a constant with convolution

Multiple answers and partial answers here, some for the more general problem of multiplying $n$ independent standard uniform random variables. For $n = 2,$ the PDF of the product $Z = XY$ is $f(z) = -...
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  • 50.7k
8 votes
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How can one construct a cumulative probability distribution function from 2 others?

The question asks for the expected time to complete both of two independent tasks. Call these times $X_1$ and $X_2$: they are random variables supported on $[0,\infty)$. Let $F_i$ be the cumulative ...
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  • 290k
8 votes

Difference between joint density and density function of sum of two independent uniform random variables

If you don't write down the support, you may not see what's going on -- but as soon as you do, it's a lot clearer. I am not able to understand the difference between the joint density function and ...
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  • 261k
8 votes

Difference between joint density and density function of sum of two independent uniform random variables

Following up on Glen_b's answer, and in an attempt to dumb it down a bit more, the following illustrations shows how the bivariate or joint pdf of $X$ and $Y$, both independent and standard uniform ...
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8 votes

Derivative of the Joint Distribution Interpretation

The first-order partial derivatives of a multivariate joint distribution function can be considered as giving the density of the differentiated variable, jointly with the cumulative probability of the ...
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  • 96.7k
8 votes

Prove 2 identical uniform's are independent by computing the joint distribution

The joint distribution of $(A^-,C^ -)=(A-B,C-B)$ is given by its density \begin{align} f(a^-,c^-)&=\int f_A(a^-+b)f_C(c^-+b)f_B(b)\,\text d b\\ &=\int_0^1 \mathbb I_{(0,1)}(a^-+b)\mathbb I_{(0,...
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  • 92.6k
8 votes
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Is there a difference between marginal likelihood and likelihood of a marginal distribution?

The problem here is that although the observations $x_1,...,x_n$ are independent conditional on $\theta$, they are not independent conditional on $\alpha$ instead, so as a general rule: $$f(\mathbf{x}...
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  • 96.7k
8 votes

Why don't we see Copula Models as much as Regression Models?

A short answer is that in practice for many applications we don't need the joint probability distributions. A cynic would say that it's also because the users don't event understand what is a joint ...
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  • 56.6k
7 votes
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Marginal, joint, and conditional distributions of a multivariate normal

Alrighty, y'all. I have an answer. Sorry it took me so long to get it posted here. School was absolutely hectic this week. Spring break is here, though, and I can type up my answer. First we need ...
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  • 331
7 votes
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X,Y univariate random variable with $F_{X,Y}(x,y)=G_1(x)G_2(y)$: are they independent?

Yes, it's true that these assumptions imply $X$ and $Y$ are independent. Simplify the notation by writing $F = F_{X,Y}$. By definition, $$F(x,y) = \Pr(X \le x, Y \le y).$$ Therefore the limit of $...
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  • 290k
7 votes

Minimum CDF of random variables

Since it says so in the title (though not repeated in the body of the question), I'm going to assume that $X$ and $Y$ are independent; otherwise, we can't say much. One of the key properties of ...
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