Graphical relationship between joint probability density, marginal probability density, and conditional probability density

Graphical relationship between joint probability density, marginal probability density, and conditional probability density

Note source: L10.3 Comments on Conditional PDFs
The relationship between joint probability density and conditional probability density

Personal understanding:A conditional probability density function image is a scaled "slice" of the joint probability density function image(under the condition that y takes a certain value)
personal understanding:All CPDF images are all scaled "slices" of the JPDF image(under the condition that y takes all values)
$$\text{conditional probability density}=\frac{\text{Joint Probability}}{\text{marginal probability density}}$$
$$f_{X|Y}(x|y)=\frac{f_{X,Y}(x,y)}{f_Y(y)}$$
The probability density function image cut out by slices does not meet the normative requirements of the probability density function (the area under the function is 1), so it needs to be scaled, and the purpose of dividing by the marginal probability is to normalize to achieve the probability density function Require

The relationship between joint probability density and marginal probability density
Personal understanding: Actuallymarginal probability density function imageActually it is"profile" of the joint probability density function image


About XXmarginal probability of$X$

$$f_X(x)={\int }_{-\infty }^{+\infty }f(x,y)dy$$
About YYMarginal probability of$Y$

$$f_Y(y)={\int }_{-\infty }^{+\infty }f(x,y)dx$$
As an example (because of the software itself, the function image is not fully displayed)

Now find$f_X(x)$
$$f_X(x)={\int }_{-\infty }^{+\infty }f(x,y)dy$$
Take all y in [0,1]

We can verify that when $y=1$时，$f(x,y)$ Whether the integral value of the lower section is the result indicated by the slash 2

$${\int }_0^{1}f(x,1)dx={\int }_0^{1}4xdx=2$$