# Tag Info

### 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 ...
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, ...
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 $B_1,\dots,B_n$. ...

### 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 ...

### 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 ...

### 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 $$E(X) = E(X| Y \le a)P(Y \le a) + E(X|Y > a) P(Y>a)$$

### 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 ...
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 ...
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, ...
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 ...
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 ...

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 ...

### 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 ...

### 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 ...

### 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 ...
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,...