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 ...
- 31.4k
18
votes
Accepted
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, ...
- 316
16
votes
Accepted
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 $B_1,\dots,B_n$. ...
- 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 ...
- 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 ...
- 451
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
$$
E(X) = E(X| Y \le a)P(Y \le a) + E(X|Y > a) P(Y>a)
$$
- 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 ...
- 131
11
votes
Accepted
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 ...
- 290k
11
votes
Accepted
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, ...
- 1,118
11
votes
Accepted
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 ...
- 290k
10
votes
Accepted
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 ...
- 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)...
- 261k
10
votes
Accepted
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 ...
- 290k
9
votes
Accepted
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.
- 231
9
votes
Accepted
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 ...
- 12.1k
9
votes
Accepted
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 ...
- 92.6k
9
votes
Accepted
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 ...
- 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. ...
- 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 ...
- 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) = -...
- 50.7k
8
votes
Accepted
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 ...
- 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 ...
- 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 ...
- 23.7k
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 ...
- 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,...
- 92.6k
8
votes
Accepted
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:
$$f(\mathbf{x}...
- 96.7k
8
votes
Why don't we see Copula Models as much as Regression Models?
A short answer is that in practice for many applications we don't need the joint probability distributions. A cynic would say that it's also because the users don't event understand what is a joint ...
- 56.6k
7
votes
Accepted
Marginal, joint, and conditional distributions of a multivariate normal
Alrighty, y'all. I have an answer. Sorry it took me so long to get it posted here. School was absolutely hectic this week. Spring break is here, though, and I can type up my answer.
First we need ...
- 331
7
votes
Accepted
X,Y univariate random variable with $F_{X,Y}(x,y)=G_1(x)G_2(y)$: are they independent?
Yes, it's true that these assumptions imply $X$ and $Y$ are independent.
Simplify the notation by writing $F = F_{X,Y}$. By definition,
$$F(x,y) = \Pr(X \le x, Y \le y).$$
Therefore the limit of $...
- 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 ...
- 22.3k
Only top scored, non community-wiki answers of a minimum length are eligible