Research on marginal probability distribution and marginal density function

The problem of edge density function derived from edge distribution

I was pretty good at mathematics in college, especially in engineering mathematical analysis. We are using the Soviet version of the analytic geometry textbook. If you have any questions, you can ask me for help.

Regarding the first question you mentioned, it is wrong to use the distribution function to derive the density function. Although most distribution functions are continuous, this does not mean that they can be derived as density functions. The Cauchy distribution is a well-known counterexample, for which a distribution function exists but a density function cannot be derived. The Cauchy distribution is used as a counterexample because it is a special case and was proposed by a famous scientist. If you are interested in this, you can check out the relevant materials for more information. However, for general functions it is indeed possible to derive the density function in this way.

I tell you why, because you draw a line to determine its area. For other places, even if the upper and lower limits of integration exist, there is no density in those places, that is, the integrand is zero, so the integration result is zero, so It can be omitted, and you only need to find where the density exists for integration.

3, first of all, you have to understand what expectations are, they are average values! Then you look at what the integral is, it is the area of ​​the enclosure! ! ! In the cross coordinate system, Fx is the height, dx is the base width, and when multiplied together they are the area of ​​a small rectangle. After adding up in this way, after calculating the so-called area, dividing it by the total length, we will get an average height. This average height is the expectation. To put it simply, it is to use a rectangle with the same base and equal height to be equivalent to an irregular trapezoid with a constant base length and varying height. I think I should basically explain it.

A simple question on the marginal density of random variables in probability theory Help!

There will be no problem if you write it this way

F(x)：=∫f(x,y)dy integration interval (﹣∞,﹢∞)

=∫6xydy (x²~1)

When x=1,f(x)=0;

2. Edge density of Y:

When 0

G(y):=∫f(x,y)dx integration interval (﹣∞, +∞)

=∫6xydx (0~y under the quadratic root)

Here F and G are two different functions, not equal to f.

1. Yes

2. Yes

3. Due to the condition of (0

4. The same as the understanding of 2, when x is a constant, f is not equal to zero when y is only between (0~y under the quadratic root), and the part where f is equal to zero can be ignored.

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