How to calculate the probability density function Interpolation

How to find the probability density function? How to find the probability density function?

1, first find the distribution function:

Y must be distributed on (1, e), X = ln(Y) obeys a uniform distribution

F(X)=P(x<=X)=X; //X obeys a uniform distribution on (0,1)

P(ln(y)<=X)=X; //substituting x=ln(y). Note that it is lowercase

P(y<=e^X)=X; // the internal condition transforms to take y as the variable

P(y<=Y)=ln(Y); // substituting X=ln(Y), and note that it is uppercase

i.e. F(Y)=P(y<=Y)=ln(Y).

2, then find the probability density:

f(y)=F'(Y)=1/Y; //probability density is the derivative of the distribution function

3, check the value of the Y variable

no overlap, no exceeding, the original solution is correct

How to solve the probability density function

Calculate as follows:

Set the random variables X~N(0,1) and Y~N(0,1) and X and Y are independent of each other, i.e., a tabular distribution with degree of freedom 2.

If X~N(0,1) then X^2~Ga(1/2,1/2)

According to the additivity of Ga distribution we get χ^2~Ga(n/2,1/2);

So X^2+Y^2~χ^2(2).

Basic types

Simply put, a random variable is a quantitative manifestation of a random event. For example, a batch of animals injected with a certain poison, the number of deaths in a certain period of time; a measurement of the amount of hemoglobin per person among a number of male healthy adults in a certain place; and so on. Some other phenomena are not directly expressed in quantity, such as the male and female sex of the population, positive or negative test results, etc., but we can stipulate that the male is 1, the female is 0, then the non-quantitative signs can also be expressed in quantity.

The quantities referred to in these examples, although they are various in their specifics, express the same situation from the mathematical point of view, which is that each variable can randomly acquire a different value, and it is impossible to predict that this variable will acquire a certain definite value before we carry out the test or measurement.